Examples of tasks on the locus of points.

Geometry (Greek geometria, from ge - Earth and metreo - measure)

branch of mathematics that studies spatial relations and forms, as well as other relations and forms similar to spatial ones in their structure.

The origin of the term "G.", which literally means "earth survey", can be explained by the following words attributed to the ancient Greek scientist Eudemus of Rhodes (4th century BC): "Geometry was discovered by the Egyptians and arose when measuring the Earth. This measurement was him necessary due to the flooding of the Nile River, which constantly washed away borders. " Already among the ancient Greeks, geodesy meant a mathematical science, while the term Geodesy was introduced for the science of measuring the Earth. Judging by the surviving fragments of ancient Egyptian writings, gravity developed not only from measurements of the earth, but also from measurements of volumes and surfaces during earthworks and construction work, and so on.

The initial concepts of gravity arose as a result of an abstraction from all properties and relations of bodies, except for the relative position and size. The first are expressed in the touch or adjoining of bodies to each other, in the fact that one body is a part of another, in the location “between”, “inside”, etc. The latter are expressed in the concepts of "more", "less", in the concept of the equality of bodies.

By the same abstraction, the concept of a geometric body arises. A geometric body is an abstraction in which only the shape and dimensions are preserved in complete abstraction from all other properties. At the same time, as is typical of mathematics in general, geometry completely abstracts itself from the indeterminacy and mobility of real shapes and sizes and considers all the relationships and forms it investigates to be absolutely precise and definite. Abstraction from the extension of bodies leads to the concepts of surfaces, lines and points. This is clearly expressed, for example, in the definitions given by Euclid: "a line is a length without width", "a surface is that which has length and width". A point without any extension is an abstraction that reflects the possibility of an unlimited reduction in all dimensions of a body, the imaginary limit of its infinite division. Then there is a general concept of a geometric figure, which is understood not only as a body, surface, line or point, but also any combination of them.

G. in its original meaning is the science of figures, the mutual arrangement and size of their parts, as well as the transformation of figures. This definition is in full agreement with the definition of geometry as the science of spatial forms and relationships. Indeed, the figure, as it is considered in G., is a spatial form; therefore, in G. they say, for example, “ball”, and not “body of spherical shape”; location and dimensions are determined by spatial relationships; Finally, transformation, as it is understood in G., is also a certain relation between two figures - the given one and the one into which it is transformed.

In the modern, more general sense, geometry encompasses a variety of mathematical theories, whose belonging to geometry is determined not only by the similarity (albeit sometimes very remote) of their subject matter with ordinary spatial forms and relationships, but also by the fact that they have historically developed and are being formed on G. in its original meaning and in their constructions proceed from the analysis, generalization, and modification of its concepts. Geography in this general sense is closely intertwined with other branches of mathematics, and its boundaries are not precise. See Generalization of Geometry and Modern Geometry.

Development of geometry. In the development of geology, four main periods can be indicated, the transitions between which signified a qualitative change in geography.

The first - the period of the birth of geometry as a mathematical science - proceeded in ancient Egypt, Babylon and Greece until about the 5th century. BC e. Primary geometric information appears at the earliest stages of the development of society. The beginnings of science should be considered the establishment of the first general laws, in this case, the dependencies between geometric quantities. This moment cannot be dated. The earliest work containing the rudiments of G. has come down to us from ancient Egypt and dates back to about the 17th century. BC e., but it is certainly not the first. Geometric information of that period was not numerous and was reduced primarily to the calculation of certain areas and volumes. They were stated in the form of rules, apparently, to a large extent of empirical origin, while the logical proofs were probably still very primitive. Greece, according to Greek historians, was transferred to Greece from Egypt in the 7th century. BC e. Here, over the course of several generations, it evolved into a coherent system. This process took place through the accumulation of new geometric knowledge, the elucidation of connections between various geometric facts, the development of methods of proof, and, finally, the formation of concepts about a figure, about a geometric sentence, and about proof.

This process has finally led to a qualitative leap. Geometry became an independent mathematical science: its systematic expositions appeared, in which its propositions were consistently proved. Since that time, the second period of development of geography begins. There are known references to systematic presentations of geology, among which is given in the 5th century. BC e. Hippocrates of Chios (See Hippocrates of Chios). They survived and played a decisive role in the future, which appeared around 300 BC. e. "Beginnings" of Euclid (See Beginnings of Euclid). Here geometries are presented in the way that they are still generally understood today, if we confine ourselves to elementary geometry (see elementary geometry); this is the science of the simplest spatial forms and relationships, developed in a logical sequence, based on clearly formulated basic provisions - axioms and basic spatial representations. Geometry developed on the same foundations (axioms), even refined and enriched both in the subject and in the methods of investigation, is called Euclidean geometry. Even in Greece, new results are added to it, new methods for determining areas and volumes arise (Archimedes, 3rd century BC), the doctrine of conic sections (Apollonius of Perga, 3rd century BC), the beginnings of trigonometry are added (Hipparchus , 2 in. BC e.) and G. on the sphere (Menelaus, 1st century AD). The decline of ancient society led to comparative stagnation in the development of gypsy, but it continued to develop in India, Central Asia, and the countries of the Arab East.

The renaissance of the sciences and arts in Europe led to a further flourishing of geography. A fundamentally new step was taken in the first half of the 17th century. R. Descartes, who introduced the method of coordinates into geometry. The method of coordinates made it possible to link geometry with the then-developing algebra and the emerging analysis. The application of the methods of these sciences in geology gave rise to analytic geography, and then differential geology. G. has moved to a qualitatively new level in comparison with G. of the ancients: it already considers much more general figures and uses essentially new methods. Since that time, the third period of development of G has begun. Analytic geometry studies the figures and transformations given by algebraic equations in rectangular coordinates, using the methods of algebra. Differential geometry, which arose in the 18th century. As a result of the work of L. Euler, H. Monge, and others, he already studies any sufficiently smooth curved lines and surfaces, their families (i.e., their continuous collections), and transformations (the concept of “differential geometry” is now often given more general meaning, which is discussed in the Modern Geometry section). Its name is associated mainly with its method, which comes from the differential calculus. By the 1st half of the 17th century. refers to the origin of projective geometry (See projective geometry) in the works of J. Desargues and B. Pascal (See Pascal). It arose from the problems of depicting bodies on a plane; its first subject is those properties of plane figures that are preserved when projecting from one plane to another from any point. The final formulation and systematic exposition of these new trends in geology were given in the 18th and early 19th centuries. Euler for analytical graphing (1748), Monge for differential graphing (1795), J. Poncelet for projective graphing (1822), and the very doctrine of geometric representation (in direct connection with the problems of drawing) was developed even earlier (1799) and brought into the system by Monge in the form of descriptive geometry (See descriptive geometry). In all these new disciplines, the foundations (axioms, initial concepts) of geometry remained unchanged, while the range of figures studied and their properties, as well as the methods used, expanded.

The fourth period in the development of geometry opens with the construction of N. I. Lobachevsky (See Lobachevsky) in 1826 a new, non-Euclidean geometry, now called Lobachevsky geometry (See Lobachevsky geometry). Independently of Lobachevsky, in 1832, J. Bolyai built the same geometry (K. Gauss developed the same ideas, but he did not publish them). The source, essence, and significance of Lobachevsky's ideas boil down to the following. In the geometry of Euclid, there is an axiom about parallels, which states: “through a point that does not lie on a given line, one can draw at most one line parallel to a given one.” Many geometers have tried to prove this axiom from other basic premises of Euclid's geometry, but without success. Lobachevsky came to the conclusion that such a proof is impossible. The statement opposite to Euclid's axiom says: "through a point that does not lie on a given line, one can draw not one, but at least two lines parallel to it." This is Lobachevsky's axiom. According to Lobachevsky, the addition of this provision to other basic provisions of G. leads to logically flawless conclusions. The system of these conclusions forms a new, non-Euclidean geometry. The merit of Lobachevsky lies in the fact that he not only expressed this idea, but actually built and comprehensively developed a new geometry, logically just as perfect and rich in conclusions as Euclidean, despite its inconsistency with the usual visual representations. Lobachevsky considered his geometry as a possible theory of spatial relations; however, it remained hypothetical until its real meaning was clarified (in 1868), and thus its full justification was given (see section Interpretations of Geometry).

The revolution in geometry brought about by Lobachevsky is in its significance not inferior to any of the revolutions in the natural sciences, and it is not for nothing that Lobachevsky was called the "Copernicus of Geometry." Three principles were outlined in his ideas, which determined the new development of geometries. The first principle is that not only Euclidean geometries are logically conceivable, but also other "geometries". The second principle is the principle of the very construction of new geometric theories by modifying and generalizing the main provisions of Euclidean G. The third principle is that the truth of a geometric theory, in the sense of correspondence to the real properties of space, can only be verified by physical research and it is possible that such research establish, in this sense, the inaccuracy of the Euclidean G. Modern physics has confirmed this. However, the mathematical accuracy of Euclidean geometry is not lost because of this, since it is determined by the logical consistency (consistency) of this G. In the same way, in relation to any geometric theory, one must distinguish between their physical and mathematical truth; the first consists in the conformity of reality verified by experience, the second in logical consistency. Lobachevsky gave, thus, a materialistic approach to the philosophy of mathematics. These general principles have played an important role not only in mathematics, but also in mathematics in general, in the development of its axiomatic method, and in the understanding of its relation to reality.

The main feature of the new period in the history of geometry, begun by Lobachevsky, is the development of new geometric theories - new "geometries" and in the corresponding generalization of the subject of geometry; the concept of various kinds of “spaces” arises (the term “space” has two meanings in science: on the one hand, it is an ordinary real space, on the other, it is an abstract “mathematical space”). At the same time, some theories took shape within Euclidean geography in the form of its special chapters and only then acquired independent significance. This is how projective, affine, conformal geometry, and others, were formed, the subject of which are the properties of figures that are preserved under appropriate (projective, affine, conformal, etc.) transformations. The notion of projective, affine, and conformal spaces arose; Euclidean geography itself began to be regarded in a certain sense as the head of projective geography. theories, like Lobachevsky's geometry, were built from the very beginning on the basis of a change and generalization of the concepts of Euclidean geometry. Thus, multidimensional geometry was created, for example; the first works related to it (G. Grassman and A. Cayley, 1844) represented a formal generalization of the usual analytic gravity from three coordinates to n. Some result of the development of all these new "geometries" was summed up in 1872 by F. Klein, indicating the general principle of their construction.

A fundamental step was taken by B. Riemann (lecture 1854, published 1867). First, he clearly formulated the generalized concept of space as a continuous collection of any homogeneous objects or phenomena (see the section Generalization of the subject of geometry). Secondly, he introduced the concept of space with any law for measuring distances in infinitesimal steps (similar to measuring the length of a line with a very small scale). From here developed the vast region of Georgia, the so-called. Riemannian geometry and its generalizations, which have found important applications in the theory of relativity, in mechanics, etc.

Another example. The state of the gas in the cylinder under the piston is determined by pressure and temperature. The totality of all possible states of a gas can therefore be represented as a two-dimensional space. The "points" of this "space" are the states of the gas; "points" differ in two "coordinates" - pressure and temperature, just as points on a plane differ in the values ​​of their coordinates. The continuous change of state is represented by a line in this space.

Further, one can imagine any material system - mechanical or physico-chemical. The totality of all possible states of this system is called "phase space". The "points" of this space are the states themselves. If the state of the system is defined n quantities, then we say that the system has n degrees of freedom. These quantities play the role of the coordinates of the point-state, as in the gas example, pressure and temperature played the role of coordinates. In accordance with this, such a phase space of the system is called n-dimensional. The change of state is represented by a line in this space; individual regions of states, distinguished by one or another feature, will be regions of the phase space, and the boundaries of the regions will be surfaces in this space. If the system has only two degrees of freedom, then its states can be represented by points on the plane. Thus, the state of a gas with pressure R and temperature T represented by a point with coordinates R and T, and the processes occurring with the gas will be represented by lines on the plane. This method of graphic representation is well known and is constantly used in physics and technology to visualize processes and their laws. But if the number of degrees of freedom is greater than 3, then a simple graphic representation (even in space) becomes impossible. Then, in order to preserve useful geometric analogies, one resorts to the concept of an abstract phase space. Thus, visual graphic methods grow into this abstract representation. The phase space method is widely used in mechanics, theoretical physics and physical chemistry. In mechanics, the motion of a mechanical system is represented by the motion of a point in its phase space. In physical chemistry, it is especially important to consider the shape and mutual adjoining of those regions of the phase space of a system of several substances that correspond to qualitatively different states. The surfaces separating these regions are the surfaces of transitions from one quality to another (melting, crystallization, etc.). In geometry itself, abstract spaces are also considered, the “points” of which are figures; this is how the "spaces" of circles, spheres, lines, etc. are defined. In mechanics and the theory of relativity, an abstract four-dimensional space is also introduced, adding time to the three spatial coordinates as the fourth coordinate. This means that events must be distinguished not only by position in space, but also in time.

Thus, it becomes clear how continuous collections of various objects, phenomena, and states can be brought under the generalized concept of space. In such a space it is possible to draw "lines" depicting continuous sequences of phenomena (states), draw "surfaces" and determine in an appropriate way "distances" between "points", thereby giving a quantitative expression of the physical concept of the degree of difference of the corresponding phenomena (states), and etc. Thus, by analogy with ordinary geometry, the "geometry" of abstract space arises; the latter may even bear little resemblance to ordinary space, being, for example, inhomogeneous in its geometric properties and finite, like an unevenly curved closed surface.

The subject of geology in a generalized sense is not only spatial forms and relations, but any forms and relations that, taken in abstraction from their content, turn out to be similar to ordinary spatial forms and relations. These space-like forms of reality are called "spaces" and "figures". Space in this sense is a continuous collection of homogeneous objects, phenomena, states that play the role of points in space, and in this collection there are relations similar to ordinary spatial relations, such as, for example, the distance between points, the equality of figures, etc. (a figure is generally a part of space). G. considers these forms of reality in abstraction from concrete content, while the study of specific forms and relations in connection with their qualitatively unique content is the subject of other sciences, and G. serves as a method for them. Any application of abstract geometry can serve as an example, even if the above application n-dimensional space in physical chemistry. G. is characterized by such an approach to the object, which consists in generalizing and transferring to new objects the usual geometric concepts and visual representations. This is exactly what is done in the above examples of the space of colors, etc. This geometric approach is not at all a pure convention, but corresponds to the very nature of phenomena. But often the same real facts can be represented analytically or geometrically, just as the same dependence can be given by an equation or a line on a graph.

One should not, however, represent the development of geometry in such a way that it only registers and describes in geometric language the forms and relationships that have already been encountered in practice, similar to spatial ones. In reality, geometry defines broad classes of new spaces and figures in them, proceeding from an analysis and generalization of the data of visual geometry and already established geometric theories. In the abstract definition, these spaces and figures appear as possible forms of reality. They, therefore, are not purely speculative constructions, but should ultimately serve as a means of research and description of real facts. Lobachevsky, creating his geometry, considered it a possible theory of spatial relations. And just as his geometry was substantiated in the sense of its logical consistency and applicability to natural phenomena, so any abstract geometric theory passes the same double test. To check the logical consistency, the method of constructing mathematical models of new spaces is essential. However, only those abstract concepts are finally rooted in science that are justified both by the construction of an artificial model and by applications, if not directly in natural science and technology, then at least in other mathematical theories through which these concepts are somehow connected with reality. The ease with which mathematicians and physicists now operate with different "spaces" has been achieved as a result of the long development of geometry in close connection with the development of mathematics as a whole and other exact sciences. It is precisely as a result of this development that the second side of geography, indicated in the general definition given at the beginning of the article, took shape and acquired great significance: the inclusion in geography of the study of forms and relations similar to forms and relations in ordinary space.

As an example of an abstract geometric theory, one can consider G. n-dimensional Euclidean space. It is constructed by a simple generalization of the main provisions of ordinary geometry, and there are several possibilities for this: one can, for example, generalize the axioms of ordinary geometry, but one can also proceed from specifying points by coordinates. With the second approach n-dimensional space is defined as a set of any element-points given by (each) n numbers x 1, x2,…, xn, located in a certain order, - the coordinates of the points. Further, the distance between the points X \u003d (x 1, x 2, ..., xn) and X"= (x’ 1, x’ 2,…, x’ n) is determined by the formula:

which is a direct generalization of the well-known formula for distance in three-dimensional space. Motion is defined as a transformation of a figure that does not change the distances between its points. Then the subject n-dimensional geometry is defined as the study of those properties of figures that do not change during movement. On this basis, the concepts of a straight line, of planes of various numbers of dimensions from two to n-1, about the ball, etc. That. a theory rich in content is emerging, in many respects similar to ordinary Euclidean geometry, but in many respects also different from it. It often happens that the results obtained for a three-dimensional space are easily transferred, with appropriate changes, to a space of any number of dimensions. For example, the theorem that among all bodies of the same volume, the ball has the smallest surface area, is read verbatim in the same way in the space of any number of dimensions [you just need to keep in mind n-dimensional volume, ( n-1)-dimensional area and n-dimensional ball, which are defined quite analogously to the corresponding concepts of ordinary gravity]. Next, in n-dimensional space, the volume of a prism is equal to the product of the base area and the height, and the volume of the pyramid is equal to such a product divided by n. Such examples could be continued. On the other hand, qualitatively new facts are also found in multidimensional spaces.

Interpretations of geometry. The same geometric theory allows different applications, different interpretations (realizations, models, or interpretations). Any application of a theory is nothing but the realization of some of its conclusions in the corresponding field of phenomena.

The possibility of different implementations is a common property of any mathematical theory. Thus, arithmetic relations are realized on the most diverse sets of objects; the same equation often describes completely different phenomena. Mathematics considers only the form of a phenomenon, abstracting from the content, and from the point of view of form, many qualitatively different phenomena often turn out to be similar. The variety of applications of mathematics and, in particular, geometry is ensured precisely by its abstract character. It is believed that a certain system of objects (a field of phenomena) provides the realization of a theory if the relations in this field of objects can be described in the language of the theory in such a way that each statement of the theory expresses one or another fact that takes place in the area under consideration. In particular, if a theory is built on the basis of a certain system of axioms, then the interpretation of this theory consists in such a comparison of its concepts with certain objects and their relations, in which the axioms are satisfied for these objects.

Euclidean G. arose as a reflection of the facts of reality. Its usual interpretation, in which stretched threads are considered straight, mechanical movement, etc., precedes gravity as a mathematical theory. The question of other interpretations was not and could not be raised until a more abstract understanding of geometry emerged. Lobachevsky created non-Euclidean geometry as a possible geometry, and then the question arose about its real interpretation. This problem was solved in 1868 by E. Beltrami, who noticed that Lobachevsky's geometry coincides with the internal geometry of surfaces of constant negative curvature, i.e., Lobachevsky's geometry theorems describe geometric facts on such surfaces (in this case, the role of straight lines is played by geodesic lines, and the role movements - bending the surface towards itself). Since, at the same time, such a surface is an object of Euclidean geometry, it turned out that Lobachevsky's geometry is interpreted in terms of Euclid's geometry. Thus, the consistency of the Lobachevsky geometry was proved, since a contradiction in it, by virtue of this interpretation, would entail a contradiction in Euclid's geometry.

Thus, the dual meaning of the interpretation of geometric theory is clarified - physical and mathematical. If we are talking about interpretation on specific objects, then we get an experimental proof of the truth of the theory (of course, with appropriate accuracy); if the objects themselves have an abstract character (like a geometric surface within the framework of Euclid's geometry), then the theory is associated with another mathematical theory, in this case with Euclidean geometry, and through it with the experimental data summarized in it. Such an interpretation of one mathematical theory by means of another has become a mathematical method of substantiating new theories, a method of proving their consistency, since a contradiction in a new theory would give rise to a contradiction in the theory in which it is interpreted. But the theory by which the interpretation is made, in turn, needs to be substantiated. Therefore, the indicated mathematical method does not remove the fact that practice remains the final criterion of truth for mathematical theories. At present, geometric theories are most often interpreted analytically; for example, points on the Lobachevsky plane can be associated with pairs of numbers X and at, straight lines - to be determined by equations, etc. This technique provides a justification for the theory because the mathematical analysis itself is justified, in the final analysis, by the vast practice of its application.

modern geometry. The formal mathematical definition of the concepts of space and figure accepted in modern mathematics proceeds from the concept of a set (see set theory). Space is defined as a set of any elements ("points") with the condition that in this set some relations are established that are similar to ordinary spatial relations. The set of colors, the set of states of the physical system, the set of continuous functions defined on the segment , etc. form spaces where the points will be colors, states, functions. More precisely, these sets are understood as spaces if only the corresponding relations are fixed in them, for example, the distance between points, and those properties and relations that are determined through them. Thus, the distance between functions can be defined as the maximum of the absolute value of their difference: max| f(x)-g(x)| . A figure is defined as an arbitrary set of points in a given space. (Sometimes space is a system of sets of elements. For example, in projective geometry it is customary to consider points, lines, and planes as equal initial geometric objects connected by “connection” relations.)

The main types of relationships that, in various combinations, lead to the whole variety of “spaces” of contemporary geometry are as follows:

1) The general relations that exist in any set are the membership and inclusion relations: a point belongs to a set, and one set is a part of another. If only these relations are taken into account, then no “geometry” is yet defined in the set, it does not become space. However, if some special figures (sets of points) are selected, then the "geometry" of space can be determined by the laws of connection of points with these figures. Such a role is played by the combination axioms in elementary, affine, and projective geometry; here the lines and planes serve as special sets.

The same principle of selecting some special sets allows us to define the concept of a topological space - a space in which "neighborhoods" of points are distinguished as special sets (with the condition that the point belongs to its own neighborhood and each point has at least one neighborhood; imposing further requirements on the neighborhoods determines one or another type of topological spaces). If any neighborhood of a given point has common points with some set, then such a point is called a point of contact of this set. Two sets can be called touching if at least one of them contains points of contact of the other; a space or figure will be continuous, or, as they say, connected if it cannot be divided into two non-contiguous parts; a transformation is continuous if it does not break contact. Thus, the concept of a topological space serves as a mathematical expression for the concept of continuity. [A topological space can also be defined by other special sets (closed, open) or directly by a tangency relation, in which any set of points is associated with its tangent points.] Topological spaces as such, sets in them, and their transformations are the subject of topology. The subject of geometry proper (to a large extent) is the study of topological spaces and figures in them, endowed with additional properties.

2) The second most important principle for determining certain spaces and their study is the introduction of coordinates. A manifold is a (connected) topological space in the neighborhood of each point of which one can introduce coordinates by putting the points of the neighborhood in a one-to-one and mutually continuous correspondence with systems from n real numbers x 1 , x 2 ,(, xn. Number n is the number of dimensions of the manifold. The spaces studied in most geometric theories are manifolds; the simplest geometric figures (segments, parts of surfaces bounded by curves, etc.) are usually pieces of manifolds. If among all the coordinate systems that can be introduced in the pieces of the manifold, coordinate systems of such a kind are distinguished that some coordinates are expressed in terms of others by differentiable (one or another number of times) or analytic functions, then we get the so-called. smooth (analytic) manifold. This concept generalizes the visual representation of a smooth surface. Smooth manifolds as such are the subject of the so-called. differential topology. In G. proper, they are endowed with additional properties. Coordinates with the accepted condition of differentiability of their transformations provide the basis for the widespread use of analytical methods - differential and integral calculus, as well as vector and tensor analysis (see Vector calculus, Tensor calculus). The totality of the theories of geology developed by these methods forms a general differential geography; its simplest case is the classical theory of smooth curves and surfaces, which are nothing but one- and two-dimensional differentiable manifolds.

3) The generalization of the concept of motion as a transformation of one figure into another leads to a general principle for defining different spaces, when a space is a set of elements (points) in which a group of one-to-one transformations of this set onto itself is given. The "geometry" of such a space consists in the study of those properties of figures that are preserved under transformations from this group. Therefore, from the point of view of such a geometry, figures can be considered "equal" if one passes into the other through a transformation from a given group. For example, Euclidean geometry studies the properties of figures that are preserved under motions, affine geometry studies the properties of figures that are preserved under affine transformations, and topology studies the properties of figures that are preserved under any one-to-one and continuous transformations. Lobachevsky geometry, projective geometries, and others are included in this scheme. In fact, this principle is combined with the introduction of coordinates. A space is defined as a smooth manifold in which transformations are defined by functions relating the coordinates of each given point and the one to which it passes (the coordinates of the image of a point are defined as functions of the coordinates of the point itself and the parameters on which the transformation depends; for example, affine transformations are defined as linear: x" i = a i1 x 1 + a i2 x 2 +…+ a in x n , i = 1, …, n). Therefore, the general apparatus for the development of such "geometries" is the theory of continuous groups of transformations. Another, essentially equivalent, point of view is possible, according to which not space transformations are specified, but coordinate transformations in it, and those properties of figures that are equally expressed in different coordinate systems are studied. This point of view has found application in the theory of relativity, which requires the same expression of physical laws in different coordinate systems, called frames of reference in physics.

4) Another general principle for the definition of spaces, indicated in 1854 by Riemann, proceeds from a generalization of the concept of distance. According to Riemann, space is a smooth manifold in which the law of measuring distances, more precisely, lengths, is set in infinitesimal steps, i.e., the differential of the length of the arc of the curve is set as a function of the coordinates of the point of the curve and their differentials. This is a generalization of the internal geometry of surfaces, defined by Gauss as the study of the properties of surfaces, which can be established by measuring the lengths of curves on it. The simplest case is represented by the so-called. Riemannian spaces in which the Pythagorean theorem holds in the infinitely small (i.e., in a neighborhood of each point, one can introduce coordinates in such a way that at this point the square of the differential of the arc length will be equal to the sum of the squares of the differentials of the coordinates; in arbitrary coordinates, it is expressed by a general positive quadratic form, see Riemannian geometries (see Riemannian geometry)). Such a space, therefore, is Euclidean in the infinitesimal, but in general it may not be Euclidean, just as a curved surface can only be reduced to a plane in the infinitesimal with the appropriate accuracy. The geometries of Euclid and Lobachevsky turn out to be a special case of this Riemannian G. The broadest generalization of the concept of distance led to the concept of a general metric space as such a set of elements in which a "metric" is given, i.e., each pair of elements is assigned a number - the distance between them, subordinate only very general conditions. This idea plays an important role in functional analysis and underlies some of the newest geometric theories, such as the intrinsic boundary of nonsmooth surfaces and the corresponding generalizations of the Riemannian boundary.

5) The combination of Riemann's idea about the definition of "geometry" in infinitely small areas of a manifold with the definition of "geometry" by means of a group of transformations led (E. Cartan, 1922-25) to the concept of a space in which transformations are given only in infinitely small areas; in other words, here the transformations establish a connection between only infinitely close pieces of the manifold: one piece is transformed into another, infinitely close one. Therefore, one speaks of spaces with a "connection" of one type or another. In particular, spaces with a "Euclidean connection" are Riemannian. Further generalizations go back to the concept of space as a smooth manifold, on which the “field” of some “object” is given in general, which can be a quadratic form, as in Riemannian geometry, a set of quantities that determine a connection, one or another tensor, etc. This also includes the recently introduced so-called. layered spaces. These concepts include, in particular, a generalization of Riemannian geometry associated with the theory of relativity, when spaces are considered where the metric is no longer given by a positive, but by an alternating quadratic form (such spaces are also called Riemannian, or pseudo-Riemannian, if they want to distinguish them from Riemannian in the original sense ). These spaces are spaces with a connection defined by the corresponding group, different from the group of Euclidean motions.

On the basis of the theory of relativity, a theory of spaces arose in which the concept of succession of points is defined, so that each point X answers set V(X) points following it. (This is a natural mathematical generalization of the sequence of events, defined by the fact that the event Y follows the event x, if X affects Y, and then Y follows X in time in any frame of reference.) Since the very assignment of sets V defines the points following x, as belonging to the set V(X), then the definition of this type of spaces turns out to be the application of the first of the principles listed above, when the "geometry" of the space is determined by the selection of special sets. Of course, while many V must be subject to the relevant conditions; in the simplest case, these are convex cones. This theory includes the theory of the corresponding pseudo-Riemannian spaces.

6) The axiomatic method in its pure form now serves either to formulate ready-made theories, or to determine the general types of spaces with distinguished special sets. If one or another type of more specific spaces is defined by formulating their properties as axioms, then either coordinates or a metric are used, etc. The consistency and thus the meaningfulness of an axiomatic theory is checked by indicating the model on which it is implemented, as was first done for geometry Lobachevsky. The model itself is built from abstract mathematical objects, so the "final justification" of any geometric theory goes into the realm of the foundations of mathematics in general, which cannot be final in the full sense, but require deepening (see Mathematics, Axiomatic method).

These principles in various combinations and variations give rise to a wide variety of geometric theories. The significance of each of them and the degree of attention to its problems are determined by the content of these problems and the results obtained, its connections with other theories of geometry, with other areas of mathematics, with exact natural science and with the problems of technology. Each given geometric theory is defined among other geometric theories, firstly, by what space or what type of space it considers. Secondly, the definition of a theory includes an indication of the figures under study. This is how the theories of polyhedra, curves, surfaces, convex bodies, etc. are distinguished. Each of these theories can develop in a particular space. For example, one can consider the theory of polyhedra in the usual Euclidean space, in n-dimensional Euclidean space, in Lobachevsky space, etc. It is possible to develop the usual theory of surfaces, projective, in Lobachevsky space, etc. Thirdly, the nature of the considered properties of the figures matters. Thus, one can study the properties of surfaces that are preserved under certain transformations; one can distinguish between the doctrine of the curvature of surfaces, the doctrine of bendings (i.e., of deformations that do not change the lengths of curves on the surface), and internal G. Finally, in the definition of a theory one can include its basic method and the nature of the formulation of problems. G. is distinguished in this way: elementary, analytical, differential; for example, one can speak of elementary or analytic geometries of Lobachevsky space. G. is distinguished “in the small,” which considers only the properties of arbitrarily small pieces of a geometric image (curve, surface, manifold), from G. “as a whole,” which, as is clear from its name, geometric images as a whole throughout their entire length. A very general distinction is made between analytic methods and methods of synthetic geometry (or strictly geometric methods); the former use the means of the corresponding calculus: differential, tensor, etc., the latter operate directly with geometric images.

Of all the variety of geometric theories, in fact, the most developed n-dimensional Euclidean geometry and Riemannian (including pseudo-Riemannian) geometry. In the first one, in particular, the theory of curves and surfaces (and hypersurfaces of different numbers of dimensions) is developed; smooth, studied in classical differential geometry; this also includes polyhedra (polyhedral surfaces). Then it is necessary to name the theory of convex bodies, which, however, in large part can be attributed to the theory of surfaces as a whole, since. a body is defined by its surface. Next is the theory of regular systems of figures, i.e., those that allow movements that transfer the entire system into itself and any of its figures into any other (see Fedorov groups (See Fedorov group)). It can be noted that a significant number of the most important results in these areas are due to Sov. geometers: a very complete development of the theory of convex surfaces and a significant development of the theory of general non-convex surfaces, various theorems on surfaces in general (the existence and uniqueness of convex surfaces with a given intrinsic metric or with a given one or another "curvature function", a theorem on the impossibility of the existence of a complete surface with curvature, everywhere less than some negative number, etc.), the study of the correct division of space, etc.

In the theory of Riemannian spaces, questions are investigated concerning the connection of their metric properties with the topological structure, the behavior of geodesic (shortest on small sections) lines in general, such as the question of the existence of closed geodesics, questions of "immersion", i.e., the realization of a given n-dimensional Riemannian space in the form n-dimensional surface in the Euclidean space of any number of dimensions, questions of pseudo-Riemannian geometry related to the general theory of relativity, and others. G.

In addition, mention should be made of algebraic geometry (see Algebraic geometry), which developed from analytic geometry and studies primarily geometric images defined by algebraic equations; it occupies a special place, because includes not only geometric, but also algebraic and arithmetic problems. There is also an extensive and important field of study of infinite-dimensional spaces, which, however, is not included in the category of heterogeneity, but is included in functional analysis, since Infinite-dimensional spaces are specifically defined as spaces whose points are certain functions. Nevertheless, in this area there are many results and problems that are of a truly geometric nature and which therefore should be attributed to G.

Geometry value. The use of Euclidean geometry is the most common phenomenon wherever areas, volumes, and so on are determined. All technology, since the shape and size of bodies play a role in it, uses Euclidean gyroscopy. Cartography, geodesy, astronomy, all graphic methods, mechanics are unthinkable without gyroscope. he could take advantage of the fact that the ellipse was studied by ancient geometers. Geometrical crystallography is a profound application of geometrical crystallography, which has served as a source and field of application for the theory of regular systems of figures (cf. Crystallography).

More abstract geometric theories are widely used in mechanics and physics, when the set of states of a system is considered as a certain space (see the section Generalization of the Subject of Geometry). So, all possible configurations (mutual arrangement of elements) of a mechanical system form a "configuration space"; the motion of the system is represented by the motion of a point in this space. The totality of all states of a physical system (in the simplest case, the positions and velocities of the material points forming the system, for example, gas molecules) is considered as the "phase space" of the system. This point of view finds, in particular, application in statistical physics (see. Statistical physics), etc.

For the first time, the concept of a multidimensional space was born in connection with mechanics as early as J. Lagrange, when three spaces. coordinates x, y, z time is formally added as the fourth t. This is how a four-dimensional “space-time” appears, where a point is determined by four coordinates x, y, z, t. Each event is characterized by these four coordinates and, abstractly, the set of all events in the world turns out to be a four-dimensional space. This view was developed in the geometric interpretation of the theory of relativity given by H. Minkowski and later in A. Einstein's construction of the general theory of relativity. In it, he used the four-dimensional Riemannian (pseudo-Riemannian) geometry. Thus, geometric theories, developed from the generalization of data from spatial experience, turned out to be a mathematical method for constructing a deeper theory of space and time. In turn, the theory of relativity gave a powerful impetus to the development of general geometric theories. Having arisen from elementary practice, geography returns to natural science and practice at a higher level as a method through a series of abstractions and generalizations.

From a geometric point of view, the space-time manifold is usually treated in the general theory of relativity as inhomogeneous of the Riemann type, but with a metric determined by a sign-changing form, reduced in an infinitesimal region to the form

dx 2 + dy 2 + dz 2 - c 2 dt 2

(with - speed of light in vacuum). Space itself, since it can be separated from time, also turns out to be non-homogeneous Riemannian. From a modern geometric point of view, it is better to look at the theory of relativity in the following way. The special theory of relativity claims that the manifold of space - time is a pseudo-Euclidean space, i.e. one in which the role of "movements" is played by transformations that preserve the quadratic form

x 2 + y 2 + z 2 - c 2 t 2

more precisely, it is a space with a group of transformations preserving the indicated quadratic form. Any formula expressing a physical law is required not to change under the transformations of the group of this space, which are the so-called Lorentz transformations. According to the general theory of relativity, the space-time manifold is non-homogeneous and only in each “infinitely small” region is reduced to pseudo-Euclidean, i.e. it is a Cartan-type space (see section Modern geometry). However, such an understanding became possible only later, because. the very concept of spaces of this type appeared after the theory of relativity and was developed under its direct influence.

In mathematics itself, the position and role of geometry are determined primarily by the fact that continuity was introduced into mathematics through it. Mathematics, as a science of the forms of reality, first of all encounters two general forms: discreteness and continuity. The account of separate (discrete) objects gives arithmetic, spaces. G. studies continuity. One of the main contradictions driving the development of mathematics is the clash between the discrete and the continuous. Even the division of continuous quantities into parts and measurement represent a comparison of the discrete and the continuous: for example, the scale is plotted along the measured segment in separate steps. The contradiction came to light. with particular clarity, when in ancient Greece (probably in the 5th century BC) the incommensurability of the side and diagonal of a square was discovered: the length of the diagonal of a square with side 1 was not expressed by any number, because the concept of an irrational number did not exist. It took a generalization of the concept of number - the creation of the concept of an irrational number (which was done only much later in India). The general theory of irrational numbers was created only in the 70s. 19th century The straight line (and with it any figure) began to be considered as a set of points. Now this point of view is dominant. However, the difficulties of set theory showed its limitations. The contradiction between the discrete and the continuous cannot be completely removed.

The general role of geometry in mathematics also lies in the fact that it is associated with precise synthetic thinking, which proceeds from spatial representations, and often makes it possible to grasp as a whole what is achieved by analysis and calculations only through a long chain of steps. Thus, geometry is characterized not only by its subject matter, but also by its method, which proceeds from visual representations and proves fruitful in solving many problems in other areas of mathematics. In turn, G. makes extensive use of their methods. Thus, one and the same mathematical problem can often be treated either analytically or geometrically, or in a combination of both methods.

In a certain sense, almost all mathematics can be regarded as developing from the interaction of algebra (originally arithmetic) and geometry, and in the sense of method, from a combination of calculations and geometric representations. This can already be seen in the concept of the totality of all real numbers as a number line connecting the arithmetic properties of numbers with continuity. Here are some highlights of G.'s influence in mathematics.

1) Along with mechanics, geometry was of decisive importance in the emergence and development of analysis. Integration comes from finding areas and volumes, begun by ancient scientists, moreover, area and volume as quantities were considered certain; no analytical definition of the integral was given until the first half of the 19th century. Drawing tangents was one of the problems that gave rise to differentiation. The graphical representation of functions played an important role in the development of the concepts of analysis and retains its importance. In the very terminology of analysis, the geometric source of its concepts is visible, as, for example, in the terms: “breaking point”, “range of change of a variable”, etc. The first course of analysis, written in 1696 by G. Lopital (See Lopital), was called: "Infinitesimal analysis for the understanding of curved lines." The theory of differential equations is mostly interpreted geometrically (integral curves, etc.). Calculus of variations It arose and develops to a large extent on the problems of geometry, and its concepts play an important role in it.

2) Complex numbers finally established themselves in mathematics at the turn of the 18th-19th centuries. only as a result of comparing them with points of the plane, i.e., by constructing a "complex plane". In the theory of functions of a complex variable, geometric methods play an essential role. The very concept of an analytic function w = f(z) of a complex variable can be defined purely geometrically: such a function is a conformal mapping of the plane z(or areas of the plane z) in the plane w. The concepts and methods of Riemannian geometry find application in the theory of functions of several complex variables.

3) The main idea of ​​functional analysis is that the functions of a given class (for example, all continuous functions defined on the interval ) are considered as points of the “functional space”, and the relations between functions are interpreted as geometric relations between the corresponding points (for example, the convergence of functions is interpreted as the convergence of points, the maximum of the absolute value of the difference of functions - as a distance, etc.). Then many questions of analysis receive a geometric treatment, which in many cases turns out to be very fruitful. In general, the representation of certain mathematical objects (functions, figures, etc.) as points of some space with the corresponding geometric interpretation of the relations of these objects is one of the most general and fruitful ideas of modern mathematics, which has penetrated almost all of its sections.

4) G. influences algebra and even arithmetic - number theory. Algebra uses, for example, the concept of a vector space. In number theory, a geometric direction has been created that makes it possible to solve many problems that are hardly amenable to the computational method. In turn, we should also note the graphic methods of calculation (see Nomography) and the geometric methods of the modern theory of calculations and computers.

5) The logical improvement and analysis of the axiomatics of a theory played a decisive role in the development of an abstract form of the axiomatic method with its complete abstraction from the nature of the objects and relations that appear in the axiomatized theory. On the basis of the same material, the concepts of consistency, completeness, and independence of axioms were developed.

On the whole, the interpenetration of geometry and other areas of mathematics is so close that often the boundaries turn out to be conditional and associated only with tradition. Only such sections as abstract algebra, mathematical logic, and some others remain almost or not at all connected with geometry.

Lit.: Major Classical Works. Euclid, Beginnings, trans. from Greek, book. 1-15, M. - L., 1948-50; Descartes R., Geometry, trans. from Latin., M. - L., 1938; Monge G., Applications of analysis to geometry, trans. from French, M. - L., 1936; Ponselet J. V., Traite des proprietes projectives des figures, Metz - R., 1822; Gauss KF, General research on curved surfaces, transl. from German, in the collection: On the foundations of geometry, M., 1956; Lobachevsky N.I., Poln. coll. soch., v. 1-3, M. - L., 1946-51; Bolai Ya., Appendix. Application,..., per. from Latin., M. - L., 1950; Riemann B., On the hypotheses underlying the foundations of geometry, trans. from German, in the collection: On the foundations of geometry, M., 1956; Klein, F., A Comparative Review of the Newest Geometric Research ("Erlangen Program"), ibid.; E. Kartan, Holonomy groups of generalized spaces, trans. from French, in the book: VIII International Competition for the Prize named after Nikolai Ivanovich Lobachevsky (1937), Kazan, 1940; Hilbert D., Foundations of Geometry, trans. from German., M. - L., 1948.

Story. Kolman E., History of mathematics in antiquity, M., 1961; Yushkevich A. P., History of mathematics in the Middle Ages, M., 1961; Vileitner G., The history of mathematics from Descartes to the middle of the 19th century, trans. from German, 2nd ed., M., 1966; Cantor M., Vorlesungen über die Geschichte der Mathematik, Bd 1-4, Lpz., 1907-08.

b) Elementary geometry. Hadamard J., Elementary geometry, trans. from French, part 1, 3rd ed., M., 1948, part 2, M., 1938; Pogorelov A. V., Elementary Geometry, Moscow, 1969.

in) Analytic geometry. Alexandrov P.S., Lectures on Analytic Geometry..., M., 1968; Pogorelov A. V., Analytical Geometry, 3rd ed., M., 1968.

e) Descriptive and projective geometry. Glagolev N. A., Descriptive geometry, 3rd ed., M. - L., 1953; Efimov N.V., Higher geometry, 4th ed., M., 1961.

e) Riemannian geometry and its generalizations. Rashevsky P.K., Riemannian geometry and tensor analysis, 2nd ed., M. - L., 1964; Norden A. P., Spaces of affine connection, M. - L., 1950; Cartan E., Geometry of Riemannian spaces, transl. from French, M. - L., 1936; Eisenhart L.P., Riemannian geometry, transl. from English, M., 1948.

Some monographs on geometry. Fedorov ES, Symmetry and structure of crystals. Basic works, M., 1949; Alexandrov A. D., Convex polyhedra, M. - L., 1950; his, Internal geometry of convex surfaces, M. - L., 1948; Pogorelov A. V., External geometry of convex surfaces, Moscow, 1969; Buseman G., Geometry of geodesics, trans. from English, M., 1962; his, Convex surfaces, trans. from English, M., 1964; E. Kartan, Moving Frame Method, Theory of Continuous Groups and Generalized Spaces, transl. from French, M. - L., 1936; Finikov S. P., Cartan's method of external forms in differential geometry, M. - L., 1948; his own, Projective-differential geometry, M. - L., 1937; his own, Theory of congruences, M. - L., 1950; Shouten I. A., Stroik D. J., Introduction to new methods of differential geometry, transl. from English, vol. 1-2, M. - L., 1939-48; Nomizu K., Lie groups and differential geometry, trans. from English, M., 1960; Milnor J., Morse Theory, trans. from English, M., 1965.

Dictionary of foreign words of the Russian language

4. Examples of problems on locus of points

1. Two wheels of radii r 1 and r 2 roll along a straight line l. Find the set of intersection points M of their common interior tangents.

Solution: Let O 1 and O 2 be the centers of wheels of radii r 1 and r 2, respectively. If M is the intersection point of internal tangents, then O 1 M: O 2 M = r 1: r 2 . From this condition it is easy to get that the distance from the point M to the line l is equal to 2r 1 r 2 /(r 1 + r 2). Therefore, all points of intersection of the common internal tangents lie on a straight line parallel to the straight line l and spaced from it by a distance 2r 1 r 2 /(r 1 + r 2).

2. Find the locus of centers of circles passing through two given points.

Solution: Let a circle with center O pass through given points A and B. Since OA = OB (as radii of one circle), point O lies on the perpendicular bisector of line segment AB. Conversely, each point O lying on the perpendicular bisector of AB is equidistant from points A and B. Hence, point O is the center of the circle passing through points A and B.

3. Sides AB and CD of quadrilateral ABCD of area S are not parallel. Find the HMT X lying inside the quadrilateral for which S ABX + S CDX = S/2.

Solution: Let O be the intersection point of lines AB and CD. Let us plot segments OK and OL on the rays OA and OD, equal to AB and CD, respectively. Then S ABX + S CDX = S KOX + S LOX ±S KXL . Therefore, the area of ​​the triangle KXL is constant, i.e. the point X lies on a line parallel to KL.

4. Points A and B are given on the plane. Find the GMT M for which the difference of the squares of the lengths of the segments AM and BM is constant.

Solution: We introduce a coordinate system by choosing point A as the origin and directing the Ox axis along the ray AB. Let point M have coordinates (x, y). Then AM 2 = x 2 + y 2 and BM 2 = (x - a) 2 + y 2 , where a = AB. Therefore AM 2 - BM 2 = 2ax - a 2 . This value is equal to k for points M with coordinates ((a 2 + k)/2a, y); all such points lie on a line perpendicular to AB.

5. Rectangle ABCD is given. Find the GMT X for which AX + BX = CX + DX.

Solution: Let l be a line passing through the midpoints of sides BC and AD. Suppose that the point X does not lie on the line l, for example, that the points A and X lie on the same side of the line l. Then AX< DX и BX < CX, а значит, AX + BX < CX + DX. Поэтому прямая l - искомое ГМТ.

6. Given two lines intersecting at a point O. Find the GMT X for which the sum of the lengths of the projections of the segments OX onto these lines is constant.

Solution: Let a and b be unit vectors parallel to given lines; x is equal to the vector x. The sum of the lengths of the projections of the vector x onto the given lines is equal to |(a,x)| + |(b,x)| = |(a±b,x)|, and the change of sign occurs on the perpendiculars erected from the point O to the given lines. Therefore, the desired GMT is a rectangle whose sides are parallel to the bisectors of the angles between the given lines, and whose vertices lie on the indicated perpendiculars.

7. Given a circle S and a point M outside it. Through the point M, all possible circles S 1 are drawn, intersecting the circle S; X - the point of intersection of the tangent at the point M to the circle S 1 with the continuation of the common chord of the circles S and S 1 . Find GMT X.

Solution: Let A and B be the intersection points of circles S and S 1 . Then XM 2 = XA . XB \u003d XO 2 - R 2, where O and R are the center and radius of the circle S. Therefore, XO 2 - XM 2 \u003d R 2, which means that the points X lie on the perpendicular to the line OM.

8. Two non-intersecting circles are given. Find the locus of points of the centers of the circles that bisect the given circles (i.e., intersect them at diametrically opposite points).

Solution: Let O 1 and O 2 be the centers of these circles, R 1 and R 2 are their radii. A circle of radius r with center X intersects the first circle at diametrically opposite points if and only if r 2 \u003d XO 1 2 + R 1 2, therefore the desired GMT consists of points X such that XO 1 2 + R 1 2 = XO 2 2 + R 2 2 , all such points of X lie on a line perpendicular to O 1 O 2 .

9. Point A is taken inside the circle. Find the locus of intersection points of tangents to the circle drawn through the ends of all possible chords containing point A.

Solution: Let O be the center of the circle, R its radius, M the intersection point of the tangents drawn through the ends of the chord containing point A, P the midpoint of this chord. Then OP * OM = R 2 and OP = OA cos f, where f = AOP. Therefore, AM 2 \u003d OM 2 + OA 2 - 2OM * OA cos f \u003d OM 2 + OA 2 - 2R 2, which means that the value of OM 2 - AM 2 \u003d 2R 2 - OA 2 is constant. Therefore, all points of M lie on a line perpendicular to OA.

10. Find the locus of points M that lie inside the rhombus ABCD and have the property that AMD + BMC = 180 o .

Solution: Let N be a point such that the vectors MN = DA. Then NAM = DMA and NBM = BMC, so AMBN is an inscribed quadrilateral. The diagonals of the inscribed quadrilateral AMBN are equal, so AM| BN or BM| AN. In the first case AMD = MAN = AMB, and in the second case BMC = MBN = BMA. If AMB = AMD, then AMB + BMC = 180 o and point M lies on diagonal AC, and if BMA = BMC, then point M lies on diagonal BD. It is also clear that if the point M lies on one of the diagonals, then AMD + BMC = 180 o .

11. a) Given a parallelogram ABCD. Prove that the quantity AX 2 + CX 2 - BX 2 - DX 2 does not depend on the choice of point X.

b) Quadrilateral ABCD is not a parallelogram. Prove that all points of X satisfying the relation AX 2 + CX 2 = BX 2 + DX 2 lie on the same straight line perpendicular to the segment connecting the midpoints of the diagonals.

Solution: Let P and Q be the midpoints of the diagonals AC and BD. Then AX 2 + CX 2 = 2PX 2 + AC 2 /2 and BX 2 + DX 2 = 2QX 2 + BD 2 /2, therefore, in problem b), the desired HMT consists of points X such that PX 2 - QX 2 = ( BD 2 - AC 2)/4, and in problem a) P = Q, so the quantity under consideration is equal to (BD 2 - AC 2)/2.

Literature

1. Pogorelov A.V. Geometry: A textbook for grades 7-9 of educational institutions. - M.: Enlightenment, 2000, p. 61.

2. Savin A.P. The method of geometric places / Optional course in mathematics: Textbook for grades 7-9 of high school. Comp. I.L. Nikolskaya. - M .: Education, 1991, p. 74.

3. Smirnova I.M., Smirnov V.A. Geometry: A textbook for grades 7-9 of educational institutions. – M.: Mnemosyne, 2005, p. 84.

4. Sharygin I.F. Geometry. Grades 7-9: Textbook for general educational institutions. – M.: Bustard, 1997, p. 76.

5. Internet resource: http://matschool2005.narod.ru/Lessons/Lesson8.htm

Informational causality of interactions (neutralization of entropy), associated with the processes of reflection of degrees of order (excitations), the possession of a universal system of space-time relations, allocate the “absolute quantum” into a phenomenal phenomenon of physical nature. It can be an unexpected material embodiment of that initial active substance, which objective idealism, ...

Q(y) of such a section is equal to, where y is assumed to be constant during integration. Integrating then Q(y) within the range of y, i.e. from c to d, we arrive at the second expression for the double integral (B). Here, integration is performed first over x and then over y. .Formulas (A) and (B) show that the calculation of the double integral is reduced to the sequential calculation of two ordinary ...

Geometry is a science that studies the spatial relationships and shapes of objects.

Euclidean geometry is a geometric theory based on a system of axioms first set forth in Euclid's Elements.

Geometry of Lobachevsky (hyperbolic geometry)- one of the non-Euclidean geometries, a geometric theory based on the same basic premises as ordinary Euclidean geometry, with the exception of the axiom of parallel lines, which is replaced by Lobachevsky's axiom of parallel lines.

A straight line bounded at one end and unbounded at the other is called a ray.

The part of a straight line bounded on both sides is called a line segment.

Injection- This is a geometric figure formed by two rays (sides of an angle) emanating from one point (the vertex of the angle). Two units of angle measurement are used: radians and degrees. An angle of 90° is called a right angle; an angle less than 90° is called an acute angle; An angle greater than 90° is called an obtuse angle.

Adjacent corners are angles that have a common vertex and a common side; the other two sides are extensions of one another. The sum of adjacent angles is 180°. Vertical angles are two angles with a common vertex, in which the sides of one are extensions of the sides of the other.

Angle bisector called a ray that bisects an angle.

Two lines are called parallel if they lie in the same plane and do not intersect, no matter how long they are continued. All lines parallel to one line are parallel to each other. All perpendiculars to the same line are parallel to each other, and vice versa, a line perpendicular to one of the parallel lines is perpendicular to the others. The length of the perpendicular segment enclosed between two parallel lines is the distance between them. When two parallel lines intersect with a third line, eight angles are formed, which are called in pairs: corresponding angles (these angles are pairwise equal); internal cross lying angles (they are equal in pairs); external cross lying angles (they are equal in pairs); interior one-sided angles (their sum is 180°); exterior one-sided angles (their sum is 180°).

Thales' theorem. When the sides of an angle are intersected by parallel lines, the sides of the angle are divided into proportional segments.

Axioms of geometry. Axiom of belonging: through any two points on a plane one can draw a straight line and, moreover, only one. Axiom of order: among any three points lying on a line, there is at most one point lying between two others.

Axiom of congruence (equality) segments and angles: if two segments (angles) are congruent to the third, then they are congruent with each other. Axiom of parallel lines: through any point lying outside a line, it is possible to draw another line parallel to the given one, and moreover, only one.

Axiom of continuity (Axiom of Archimedes): for any two segments AB and CD, there is a finite set of points A1, A2, …, An lying on the line AB, such that the segments AA1, A1A2, …, An-1An are congruent to the segment CD, and the point B lies between A and An.

A flat figure formed by a closed chain of segments is called a polygon.
Depending on the number of angles, a polygon can be a triangle, a quadrilateral, a pentagon, a hexagon, etc. The sum of the lengths is called the perimeter and is denoted by p.
If all the diagonals lie inside the polygon, it is called convex. The sum of the interior angles of a convex polygon is 180°*(n-2), where n is the number of angles (or sides) of the polygon.

Triangle is a polygon with three sides (or three corners). If all three angles are acute, then it is an acute triangle. If one of the angles is right, then it is a right triangle; sides forming a right angle are called legs; the side opposite the right angle is called the hypotenuse. If one of the angles is obtuse, then it is an obtuse triangle. A triangle is isosceles if two of its sides are equal. A triangle is equilateral if all its sides are equal.

In a right triangle, the following relations are true:

Area of ​​a right triangle:

Radius of inscribed circle:

In an arbitrary triangle:

A circle can be inscribed in any regular polygon and a circle can be described around it:

where a is the side, n is the number of sides of the polygon, R is the radius of the circumscribed circle, r is the radius of the inscribed circle (the apothem of a regular polygon).

Area of ​​a regular polygon:

The lengths of the sides and diagonals are related by the formula:

Basic properties of triangles:

• opposite the larger side lies a larger angle and vice versa;
• opposite equal sides are equal angles and vice versa;
• the sum of the angles of a triangle is 180°;
• continuing one of the sides of the triangle, we get the external angle. The external angle of a triangle is equal to the sum of the internal angles not adjacent to it;
• Any side of a triangle is less than the sum of the other two sides and greater than their difference.

Signs of equality of triangles: triangles are congruent if they are equal:

• two sides and the angle between them;
• two corners and the side adjacent to them;
• three sides.

Right Triangle Equality Tests: Two right triangles are congruent if one of the following conditions is true:

• their legs are equal;
• the leg and hypotenuse of one triangle are equal to the leg and hypotenuse of the other;
• the hypotenuse and acute angle of one triangle are equal to the hypotenuse and acute angle of the other;
• the leg and the adjacent acute angle of one triangle are equal to the leg and the adjacent acute angle of the other;
• the leg and opposite acute angle of one triangle are equal to the leg and opposite acute angle of the other.

The height of a triangle is the perpendicular dropped from any vertex to the opposite side (or its extension). This side is called the base of the triangle. The three heights of a triangle always intersect at one point, called the triangle's orthocenter. The orthocenter of an acute triangle is located inside the triangle, and the orthocenter of an obtuse triangle is outside; The orthocenter of a right triangle coincides with the vertex of the right angle.

The formula for the height of a triangle is:

Median is a line segment that connects any vertex of a triangle with the midpoint of the opposite side. The three medians of a triangle intersect at one point, which always lies inside the triangle and is its center of gravity. This point divides each median 2:1 from the top.

Bisector- this is a segment of the bisector of the angle from the vertex to the point of intersection with the opposite side. The three bisectors of a triangle intersect at one point, which always lies inside the triangle and is the center of the inscribed circle. The bisector divides the opposite side into parts proportional to the adjacent sides.
The formula for the bisector of a triangle is:

Median perpendicular is a perpendicular drawn from the midpoint of the segment (side). The three median perpendiculars of a triangle intersect at one point, which is the center of the circumscribed circle. In an acute triangle, this point lies inside the triangle; in obtuse - outside; in a rectangular one - in the middle of the hypotenuse. The orthocenter, center of gravity, center of the circumcircle and center of the inscribed circle coincide only in an equilateral triangle.

Pythagorean theorem. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs: c2 = a2 + b2.

In the general case (for an arbitrary triangle) we have: c2=a2+b2–2?a?b?cosC, where C is the angle between sides a and b.

quadrilateral- a figure formed by four points (vertices), no three of which lie on the same straight line, and four segments (sides) connecting them in series, which should not intersect.

Parallelogram is a quadrilateral whose opposite sides are pairwise parallel. Any two opposite sides of a parallelogram are called its bases, and the distance between them is called its height.

Parallelogram properties:

• opposite sides of a parallelogram are equal;
• opposite angles of a parallelogram are equal;
• the diagonals of a parallelogram are divided in half at the point of their intersection;
• the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its four sides.

Parallelogram area:

Radius of a circle inscribed in a parallelogram:

Rectangle is a parallelogram with all angles equal to 90°.

Basic properties of a rectangle.
The sides of a rectangle are also its heights.
The diagonals of the rectangle are equal: AC = BD.

The square of the diagonal of a rectangle is equal to the sum of the squares of its sides (according to the Pythagorean theorem).

Rectangle area: S=ab.

Rectangle Diameter:

Radius of a circle circumscribed about a rectangle:

A rhombus is a parallelogram in which all sides are equal. The diagonals of the rhombus are mutually perpendicular and bisect their angles.

The area of ​​a rhombus is expressed in terms of diagonals:

A square is a parallelogram with right angles and equal sides. A square is a special case of a rectangle and a rhombus at the same time, therefore, it has all of their properties listed above.

Square area:

Radius of a circle circumscribed about a square:

Radius of a circle inscribed in a square:

Square diagonal:

Trapeze is a quadrilateral with two opposite sides parallel. The parallel sides are called the bases of the trapezoid, and the other two are called the sides. The distance between the bases is the height. The segment connecting the midpoints of the sides is called the midline of the trapezium. The midline of a trapezoid is half the sum of the bases and parallel to them. A trapezoid with equal sides is called an isosceles trapezoid. In an isosceles trapezoid, the angles at each base are equal.

Trapezium area: , where a and b are the bases, h is the height.

Middle line of the triangle is a line segment that connects the midpoints of the sides of the triangle. The midline of a triangle is equal to half of its base and parallel to it. This property follows from the property of the trapezoid, since the triangle can be considered as a case of degeneracy of the trapezoid, when one of its bases becomes a point.

Similarity of plane figures. If you change all the dimensions of a flat figure the same number of times (similarity ratio), then the old and new figures are called similar. Two polygons are similar if their angles are equal and their sides are proportional.

Signs of similarity of triangles. Two triangles are similar if:

• all their corresponding angles are equal (two angles are enough);
• all their sides are proportional;
• two sides of one triangle are proportional to two sides of the other, and the angles included between these sides are equal.

The areas of similar figures are proportional to the squares of their similar lines (eg, sides, diameters).

Locus of points is the set of all points that satisfy certain given conditions.

Circle- This is the locus of points on a plane equidistant from one point, called the center of the circle. The segment connecting the center of the circle with any of its points is called the radius and is denoted - r. The part of the plane bounded by a circle is called a circle. Part of a circle is called an arc. A straight line passing through two points of a circle is called a secant, and its segment lying inside the circle is called a chord. The chord passing through the center of the circle is called the diameter and is denoted d. The diameter is the largest chord, equal in magnitude to two radii: d = 2r.

Where a is the real, b is the imaginary semiaxis.

Equation of a plane in space:
Ax + By + Cz + D = 0,
where x, y, z are rectangular coordinates of a variable point of the plane, A, B, C are constant numbers.
A straight line passing through a point of a circle perpendicular to the radius drawn to this point is called a tangent. This point is called the point of contact.

Tangent properties:

• the tangent to the circle is perpendicular to the radius drawn to the point of contact;
• from a point outside the circle, two tangents can be drawn to the same circle; their segments are equal.

Segment- this is the part of the circle bounded by an arc and the corresponding chord. The length of the perpendicular drawn from the middle of the chord to the intersection with the arc is called the height of the segment.

Sector- this is a part of a circle bounded by an arc and two radii drawn to the ends of this arc.

Angles in a circle. A central angle is an angle formed by two radii. An inscribed angle is the angle formed by two chords drawn from their common point. The described angle is the angle formed by two tangents drawn from one common point.

This formula is the basis for determining the radian measurement of angles. The radian measure of any angle is the ratio of the length of an arc drawn by an arbitrary radius and enclosed between the sides of this angle to its radius.

Relations between the elements of the circle.

The inscribed angle is equal to half the central angle based on the same arc. Therefore, all inscribed angles based on the same arc are equal. And since the central angle contains the same number of degrees as its arc, any inscribed angle is measured by half the arc on which it rests.

All inscribed angles based on a semicircle are right angles.

The angle formed by two chords is measured by half the sum of the arcs enclosed between its sides.

The angle formed by two secants is measured by the half-difference of the arcs enclosed between its sides.

The angle formed by a tangent and a chord is measured by half the arc enclosed within it.

The angle formed by a tangent and a secant is measured by the half-difference of the arcs enclosed between its sides.

The described angle, formed by two tangents, is measured by the half-difference of the arcs enclosed between its sides.

The products of segments of chords into which they are divided by the intersection point are equal.

The square of a tangent is equal to the product of the secant and its outer part.

A chord perpendicular to the diameter is bisected at their point of intersection.

A polygon is called inscribed in a circle, the vertices of which are located on a circle. A polygon circumscribed near a circle is a polygon whose sides are tangent to the circle. Accordingly, a circle passing through the vertices of a polygon is called circumscribed near the polygon; a circle for which the sides of a polygon are tangent is called an inscribed circle. For an arbitrary polygon, it is impossible to inscribe in it and describe a circle around it. For a triangle, this possibility always exists.

A circle can be inscribed in a quadrilateral if the sums of its opposite sides are equal. For parallelograms, this is only possible for a rhombus (square). The center of the inscribed circle is located at the intersection point of the diagonals. A circle can be circumscribed about a quadrilateral if the sum of its opposite angles is 180°. For parallelograms, this is only possible for a rectangle (square). The center of the circumscribed circle lies at the intersection point of the diagonals. A circle can be described around a trapezoid if it is isosceles. A regular polygon is a polygon with equal sides and angles.

A regular quadrilateral is a square; right triangle is an equilateral triangle. Each corner of a regular polygon is equal to 180°(n - 2)/n, where n is the number of its corners. Inside a regular polygon there is a point O, equidistant from all its vertices, which is called the center of a regular polygon. The center of a regular polygon is also equidistant from all its sides. A circle can be inscribed in a regular polygon and a circle can be circumscribed around it. The centers of the inscribed and circumscribed circles coincide with the center of a regular polygon. The radius of the circumscribed circle is the radius of a regular polygon, and the radius of the inscribed circle is its apothem.

Basic axioms of stereometry.

Whatever the plane, there are points that belong to this plane and points that do not.

If two different planes have a common point, then they intersect along a straight line passing through this point.

If two distinct lines have a common point, then one and only one plane can be drawn through them.

Through three points lying on one straight line, one can draw an infinite number of planes, which in this case form a bundle of planes. The straight line through which all planes of the beam pass is called the axis of the beam. Through any line and a point outside this line, one and only one plane can be drawn. Through two lines it is not always possible to draw a plane, then these lines are called skew.

Crossing lines do not intersect, no matter how long they are continued, but they are not parallel lines, since they do not lie in the same plane. Only parallel lines are non-intersecting lines through which a plane can be drawn. The difference between skew and parallel lines is that parallel lines have the same direction, but skew lines do not. Through two intersecting lines, one and only one plane can always be drawn. The distance between two skew lines is the length of the segment connecting the nearest points located on the skew lines. Non-intersecting planes are called parallel planes. A plane and a line either intersect (at one point) or they don't. In the latter case, the line and plane are said to be parallel to each other.

A perpendicular dropped from a point to a plane is a line segment connecting the given point with a point on the plane and running on a straight line perpendicular to the plane.

The projection of a point onto a plane is the base of the perpendicular dropped from the point onto the plane. The projection of a segment onto the plane P is a segment whose ends are the projections of the points of this segment.

A dihedral angle is a figure formed by two half-planes with a common straight line bounding them. The half-planes are called faces, and the straight line bounding them is called the edge of the dihedral angle. The plane perpendicular to the edge gives an angle at its intersection with the half-planes called the linear angle of the dihedral angle. A dihedral angle is measured by its linear angle.

polyhedral angle. If through a point we draw a set of planes that successively intersect each other along straight lines, then we get a figure called a polyhedral angle. The planes forming a polyhedral angle are called its faces; the lines along which the faces intersect in succession are called the edges of the polyhedral angle. The minimum number of faces of a polyhedral angle is three.

Parallel planes are cut out on the edges of a polyhedral angle, proportional segments and form similar polygons.

Signs of parallelism of a straight line and a plane.

If a line lying outside a plane is parallel to any line lying in that plane, then it is parallel to that plane.

If a line and a plane are perpendicular to the same line, then they are parallel.

Signs of parallel planes:

• If two intersecting lines in one plane are respectively parallel to two intersecting lines in another plane, then these planes are parallel.
• If two planes are perpendicular to the same line, then they are parallel.
• Signs of perpendicularity of a straight line and a plane.
• If a line is perpendicular to two intersecting lines lying in a plane, then it is perpendicular to that plane.
• If a plane is perpendicular to one of the parallel lines, then it is also perpendicular to the other.

A straight line that intersects a plane and is not perpendicular to it is called oblique to the plane.

Three perpendiculars theorem

A straight line lying in a plane and perpendicular to the projection of an oblique plane to this plane is also perpendicular to the oblique itself.

Signs of parallel lines in space:

• If two lines are perpendicular to the same plane, then they are parallel.
• If one of the intersecting planes contains a line parallel to another plane, then it is parallel to the line of intersection of the planes.

Equation of a straight line on a plane in a rectangular coordinate system xy:
ax + bx + c = 0, where a, b, c are constant numbers, x and y are the coordinates of the variable point M(x,y) on the line.

Signs of parallel lines:

A sign of perpendicularity of planes: if a plane passes through a line perpendicular to another plane, then these planes are perpendicular.

Theorem on a common perpendicular to two skew lines. For any two intersecting lines, there is only one common perpendicular.

Polyhedron- this is a body, the boundary of which consists of pieces of planes (polygons). These polygons are called faces, their sides are called edges, their vertices are the vertices of the polyhedron. The segments connecting two vertices and not lying on the same face are called the diagonals of the polyhedron. A polyhedron is convex if all its diagonals are inside it.

Cube- three-dimensional figure with six equal faces.

Volume and surface area of ​​a cube:

A prism is a polyhedron whose two faces (the bases of the prism) are equal polygons with respectively parallel sides, and the remaining faces are parallelograms.

The segments connecting the corresponding vertices are called side edges. The height of a prism is any perpendicular dropped from any point of the base to the plane of the other base. Depending on the shape of the polygon lying at the base, the prism can be, respectively, triangular, quadrangular, pentagonal, hexagonal, etc. If the side edges of the prism are perpendicular to the plane of the base, then such a prism is called a straight line; otherwise, it is an oblique prism. If a regular polygon lies at the base of a straight prism, then such a prism is also called regular. The diagonal of a prism is a segment connecting two vertices of the prism that do not belong to the same face.

The lateral surface area of ​​a straight prism:
S side \u003d P * H, where P is the perimeter of the base, and H is the height.

Parallelepiped is a prism whose bases are parallelograms. Thus, the parallelepiped has six faces, and all of them are parallelograms. Opposite faces are pairwise equal and parallel. The parallelepiped has four diagonals; they all intersect at one point and divide in half at it.

If the four side faces of a parallelepiped are rectangles, then it is called straight. A right parallelepiped, in which all six faces are rectangles, is called rectangular. The diagonal of a rectangular parallelepiped d and its edges a, b, c are related by the relation d2 = a2 + b2 + c2. A rectangular parallelepiped, all of whose faces are squares, is called a cube. All edges of a cube are equal.

Volume and surface area of ​​a rectangular parallelepiped:
V = a*b*c, S total = 2(ab + ac + bc).

Pyramid is a polyhedron in which one face (the base of the pyramid) is an arbitrary polygon, and the remaining faces (side faces) are triangles with a common vertex, called the top of the pyramid. The perpendicular dropped from the top of the pyramid to its base is called the height of the pyramid. Depending on the shape of the polygon lying at the base, the pyramid can be, respectively, triangular, quadrangular, pentagonal, hexagonal, etc. A triangular pyramid is a tetrahedron, a quadrangular pyramid is a pentahedron, etc. A pyramid is called regular if the base lies polygon, and its height falls to the center of the base. All side edges of a regular pyramid are equal; all side faces are isosceles triangles. The height of the side face is called the apothem of a regular pyramid.

If we draw a section parallel to the base of the pyramid, then the body enclosed between these planes and the side surface is called a truncated pyramid. Parallel faces are called bases; the distance between them is the height. A truncated pyramid is called correct if the pyramid from which it was obtained is correct. All lateral faces of a regular truncated pyramid are equal isosceles trapezoids.

Lateral surface area of ​​a regular pyramid:
, where P is the perimeter of the base; h is the height of the side face (the apothem of a regular pyramid).

The volume of the truncated pyramid:

Lateral surface area of ​​a regular truncated pyramid:
,
where P and P' are the perimeters of the bases; h is the height of the side face (the apothem of a regular truncated pyramid).

A cylindrical surface is formed by moving a straight line that retains its direction and intersects with a given line (curve). This line is called the guideline. The straight lines corresponding to the different positions of the straight line as it moves are called generators of the cylindrical surface.

A cylinder is a body bounded by a cylindrical surface with a closed guide and two parallel planes. Parts of these planes are called the bases of the cylinder. The distance between the bases is the height of the cylinder. A cylinder is straight if its generators are perpendicular to the base; otherwise the cylinder is inclined. A cylinder is called circular if its base is a circle. If a cylinder is both straight and circular, then it is called round. A prism is a special case of a cylinder.

Volume, area of ​​lateral and full surfaces of the cylinder:
,
where R is the radius of the bases; H is the height of the cylinder.

Cylindrical sections of the lateral surface of a circular cylinder.

Sections parallel to the base are circles of the same radius.

Sections parallel to the generators of the cylinder are pairs of parallel lines.

Sections that are not parallel to either the base or the generators are ellipses.

A conical surface is formed when a straight line moves, passing all the time through a fixed point, and intersecting a given line, called a guide. The lines corresponding to the various positions of the line as it moves are called generatrices of the conical surface; point is its top. The conical surface consists of two parts: one is described by a ray, the other by its continuation.

Usually, one of its parts is considered as a conical surface.

Cone- this is a body bounded by one of the parts of a conical surface with a closed guide and a plane intersecting the conical surface that does not pass through the vertex.

The part of this plane located inside the conical surface is called the base of the cone. The perpendicular dropped from the top to the base is called the height of the cone.

The pyramid is a special case of a cone. A cone is called circular if its base is a circle. The straight line connecting the top of the cone with the center of the base is called the axis of the cone. If the height of a circular cone coincides with its axis, then such a cone is called circular.

Volume, area of ​​the lateral and full surfaces of the cone:
,
where r is the radius; Sosn - area; P is the circumference of the base; L is the length of the generatrix; H is the height of the cone.

Volume and area of ​​the lateral surface of a truncated cone:

Conic sections.

Sections of a circular cone parallel to its base are circles.

A section that intersects only one part of a circular cone and is not parallel to any of its generators is an ellipse.

A section that intersects only one part of a circular cone and is parallel to one of its generators is a parabola.

A section that intersects both parts of a circular cone is generally a hyperbola consisting of two branches. In particular, if this section passes through the axis of the cone, then we obtain a pair of intersecting lines (forming a cone).

spherical surface- this is the locus of points in space, equidistant from one point, which is called the center of a spherical surface.

Ball (sphere) is a body bounded by a spherical surface. You can get a ball by rotating a semicircle (or circle) around the diameter. All plane sections of a sphere are circles. The largest circle lies in the section passing through the center of the ball, and is called the great circle. Its radius is equal to the radius of the sphere. Any two great circles intersect in the diameter of the ball. This diameter is also the diameter of the intersecting great circles. Through two points of a spherical surface located at the ends of the same diameter, one can draw an infinite number of great circles.

The volume of a sphere is one and a half times less than the volume of the cylinder described around it, and the surface of the ball is one and a half times less than the total surface of the same cylinder.

The equation of a sphere in a rectangular coordinate system is:
(x-x0)+(y-y)2+ (z-z0)=R2,
here x, y, z are the coordinates of a variable point on the sphere;
x0, y0, z0 - coordinates of the center;
R is the radius of the sphere.

Volume of a sphere and area of ​​a sphere:

The volume of the spherical segment and the area of ​​the segmented surface:
,
where h is the height of the spherical segment.

Volume and total surface area of ​​the spherical sector:
,
where R is the radius of the ball; h is the height of the spherical segment.

Volume and total surface area of ​​the spherical layer:
,
where h is the height; r1 and r2 are the radii of the bases of the spherical layer.

Volume and surface area of ​​a torus:
,
where r is the radius of the circle; R is the distance from the center of the circle to the axis of rotation.

Average curvature of the surface S at point A0:

ball parts. Part of the ball (sphere), cut off from it by any plane, is called a spherical (spherical) segment. The circle is called the base of the spherical segment. The segment of the perpendicular drawn from the center of the circle to the intersection with the spherical surface is called the height of the spherical segment. The part of the sphere enclosed between two parallel planes intersecting the spherical surface is called the spherical layer; the curved surface of a spherical layer is called a spherical belt (zone). The distance between the bases of the spherical belt is its height. The part of the ball bounded by the curved surface of a spherical segment and the conical surface, the base of which is the base of the segment, and the apex is the center of the ball, is called a spherical sector.

Symmetry.

Mirror symmetry. A geometric figure is said to be symmetric with respect to the plane S if for each point E of this figure a point E' of the same figure can be found, so that the segment EE' is perpendicular to the plane S and is divided by this plane in half. The plane S is called the plane of symmetry. Symmetrical figures, objects and bodies are not equal to each other in the narrow sense of the word, they are called mirror equal.

central symmetry. A geometric figure is said to be symmetric with respect to the center C if for each point A of this figure a point E of the same figure can be found, so that the segment AE passes through the center C and is bisected at this point. Point C in this case is called the center of symmetry.

rotation symmetry. A body has rotational symmetry if, when rotated through an angle of 360° / n (n is an integer) around some straight line AB (axis of symmetry), it completely coincides with its initial position. For n=2 we have axial symmetry.

Examples of types of symmetry. A ball (sphere) has both central and mirror symmetry and rotational symmetry. The center of symmetry is the center of the ball; the plane of symmetry is the plane of any great circle; the axis of symmetry is the diameter of the ball.

The round cone is axially symmetrical; the axis of symmetry is the axis of the cone.

A straight prism has mirror symmetry. The plane of symmetry is parallel to its bases and is located at the same distance between them.

Symmetry of plane figures.

Mirror axis symmetry. If a plane figure is symmetrical with respect to a plane (which is possible only if the plane figure is perpendicular to that plane), then the line along which these planes intersect is the axis of symmetry of the second order of this figure. In this case, the figure is called mirror-symmetrical.

central symmetry. If a plane figure has an axis of symmetry of the second order, perpendicular to the plane of the figure, then the point at which the line and the plane of the figure intersect is the center of symmetry.

Examples of symmetry of plane figures.

The parallelogram has only central symmetry. Its center of symmetry is the intersection point of the diagonals.
An isosceles trapezoid has only axial symmetry. Its axis of symmetry is a perpendicular drawn through the midpoints of the bases of the trapezoid.

The rhombus has both central and axial symmetry. Its axis of symmetry is any of its diagonals; the center of symmetry is the point of their intersection.

The locus of points (hereinafter referred to as GMT) is a plane figure consisting of points with a certain property, and not containing a single point that does not have this property.

We will consider only those HMTs that can be constructed using a compass and straightedge.

Let us consider HMT on the plane, which have the simplest and most frequently expressed properties:

1) HMT, spaced at a given distance r from a given point O, is a circle centered at point O of radius r.

2) GMT of points A and B equidistant from two given points is a straight line perpendicular to the segment AB and passing through its middle.

3) GMT equidistant from two given intersecting lines, there is a pair of mutually perpendicular lines passing through the intersection point and dividing the angles between the given lines in half.

4) GMT, spaced at the same distance h from a straight line, there are two straight lines parallel to this straight line and located on opposite sides of it at a given distance h.

5) The locus of centers of circles tangent to a given line m at a given point M on it is a perpendicular to AB at the point M (except for the point M).

6) The locus of centers of circles tangent to a given circle at a given point M on it is a straight line passing through the point M and the center of the given circle (except for the points M and O).

7) HMT, of which this segment is visible at a given angle, is two arcs of circles described on a given segment and enclosing a given angle.

8) GMT, the distances from which to two given points A and B are in the ratio m: n, is a circle (called the circle of Apollonius).

9) The locus of the midpoints of chords drawn from one point of a circle is a circle built on a segment connecting a given point with the center of a given circle, as on a diameter.

10) The locus of vertices of triangles equal in size to a given one and having a common base is two straight lines parallel to the base and passing through the vertex of the given triangle and symmetrical to it with respect to the line containing the base.

Let us give examples of finding GMT.

EXAMPLE 2.Find GMT, which are the midpoints of chords,drawn from one point of the given circle(GMT No. 9).

Decision . Let a circle with center O be given and point A be chosen on this circle from which chords are drawn. Let us show that the desired HMT is a circle built on AO as a diameter (except for point A) (Fig. 3).

Let AB be a chord and M its midpoint. Let's connect M and O. Then MO ^ AB (the radius dividing the chord in half is perpendicular to this chord). But, then RAMO = 90 0 . So M belongs to a circle with a diameter of AO (GMT No. 7). Because this circle passes through the point O, then O belongs to our GMT.

Conversely, let M belong to our GMT. Then, drawing the chord AB through M and connecting M and O, we get that РАМО = 90 0 , i.e. MO ^ AB, and, therefore, M is the middle of the chord AB. If M coincides with O, then O is the midpoint of AC.

Often the coordinate method allows you to find the GMT.

EXAMPLE 3.Find the GMT, the distance from which to two given points A and B are in the given ratio m: n (m ≠ n).

Decision . We choose a rectangular coordinate system so that points A and B are located on the Ox axis symmetrically with respect to the origin of coordinates, and the Oy axis passes through the middle of AB (Fig. 4). We set AB = 2a. Then point A has coordinates A (a, 0), point B has coordinates B (-a, 0). Let C belong to our HMT, coordinates C(x, y) and CB/CA = m/n. But Means

(*)

Let's change our equation. We have

Bodies differ from each other in weight, color, density, hardness, space they occupy, etc.

These signs are called properties of bodies.

Bodies with these properties are called physical bodies.

Between these properties, the property of the body called length.

Length there is the property of a body to occupy a certain place in space.

It is called the geometric property of the body. This property determines the shape and size of the body.

A body that has only one extension property is called a geometric body. Considering a geometric body, pay attention only to its shape and size.

The remaining properties of the body are called physical.

geometric body there is space occupied by the physical body.

The geometric body is limited on all sides. It is separated from the rest of the space by the surface of the body. To express this, they say that

Surface there is body limit.

One surface is separated from the other by a line. The line defines the surface, so the line is called the boundary of the surface.

Line there is surface limit.

The end of a line is called a dot. A point delimits and separates one line from another, which is why a point is called a line boundary.

Dot there is line limit.

Figure 1 shows a body in the form of a box closed on all sides. It is bounded by six sides that form the surface of the box. Each side of the box can be viewed as a separate surface. These sides are separated from each other by 12 lines that form the edges of the box. The lines are separated from each other by 8 points that make up the corners of the box.

Bodies, surfaces and lines are not the same size. This means that they occupy an unequal space, or an unequal extent.

body volume. The value of a geometric body is called the volume or capacity of the body.

surface area. The surface area is called the area.

Line length. The length of the line is called the length.

Length, area and volume are heterogeneous quantities. They are measured in different units and used for different purposes. To find the distance of two objects, the width of the arm, the depth of the well, the height of the tower, determine the length of the line. For this, only one measurement is made, that is, a measurement is made in one direction. When measuring, resort to units of length. These units of length are called versts, sazhens, arshins, feet, meters, etc. The unit of length has one dimension, which is why they say that

Lines have one dimension. Lines have neither width nor thickness. They are the same length.

To have an idea about the size of the picture, you need to know its length and width. Length and width give an idea of ​​the area of ​​the picture. To determine the area, it became necessary to make two measurements, or measure the picture in two directions. To determine the size of the area, units of area are used. A square is taken as a unit of area, the sides of which have a certain unit of length. Units of area are called square miles, square versts, square feet, and so on. A square verst is the area of ​​a square whose each side is equal to a verst, and so on. A unit of area has two dimensions: length and width. Since surfaces are measured in units of area, in this sense they say that

Surfaces have two dimensions. Surfaces have no thickness. They can only have length and width.

To have an idea about the capacity of a room or box, you need to know their volumes. To do this, you need to know the length, width and height of the room, that is, make three measurements or measure it in three directions. Volumes are measured in units of volume. A cube is taken as a unit of volume, each side of which is equal to one. Volume units have three dimensions: length, width, and height. Since volumes are measured in units of volumes, we say that

Bodies have three dimensions.

Units of volume are called cubic versts, cubic feet, etc. Depending on the length of the side of the cube.

A point has no length, no width, no height, or a point has no dimension.

geometric extensions. Lines, surfaces, and solids are called geometric extensions.

Geometry is the science of the properties and measurement of geometric extensions.

Geometry is the science of space. It sets out a set of necessary relationships related to the nature of space.

Formation of geometric extents by movement

A line can be viewed in the same way as a trace left by the movement of a point, a surface as a trace left by the movement of a line, and a body as a trace left by the movement of a surface. Other definitions of line, surface, and solid are based on these considerations.

Line is the locus of the moving point.

Surface is the locus of the moving line.

Body is the locus of the moving surface.

All objects considered in nature have three dimensions. There are no points, no lines, no surfaces in it, but only bodies exist. However, in geometry, points, lines, and surfaces are considered separately from bodies. At the same time, a very thin shell of the body gives us some approximate visual representation of the surface, a very thin thread or hair gives us a visual representation of the line, and the end of the thread about the point.

lines

Lines are divided into straight lines, broken lines and curves.

is the shortest distance between two points.

A tightly stretched thin thread gives some visual representation of a straight line.

Any line is denoted by letters placed at its points. Drawing 2 shows a straight line AB. In every straight line, attention is drawn to its direction and value.

The direction of a straight line is determined by its position.

there is a series and continuous connection of several straight lines having different directions.

The broken line ABCD (fig. 3) is made up of straight lines AB, BC, CD, which do not have the same direction.

there is one that cannot be composed of straight lines.

The line shown in Fig. 4, will be a curved line.

A line composed of straight lines and curves is sometimes called a compound line.

Drawing (4, a) represents such a composite line.

surfaces

Surfaces are divided into straight or flat and curved. A flat surface is called a plane.

Plane. A surface is called a plane when every straight line drawn through every two points of the surface lies on it with all its points.

Curve surface there is one that cannot be composed of planes.

A straight line drawn between any two points of a curved surface does not fit on it with all its intermediate points.

Some visual representation of the plane is given by the surface of a well-polished mirror or the surface of stagnant water. An example of curved surfaces is the surface of a billiard ball.

Sections of geometry

Geometry is divided into planimetry and solid geometry.

Planimetry studies the property of geometric extensions considered on the plane.

Stereometry studies the properties of such geometric extensions that cannot be represented in one plane.

Planimetry is called geometry on a plane, stereometry - geometry in space.

Geometry is further divided into primary and higher. In the present work, only the initial geometry is presented.

Different Forms of Expression of Geometric Truths

Geometric truths are expressed in the form of axioms, theorems, lemmas, and problems or problems.

Axiom there is truth, but its evidence does not require proof.

Examples of truths that do not require proof are the following axioms:

The whole is equal to the sum of its parts.

The whole is greater than its part. The parts are smaller than the whole.

Two quantities equal to the same third are equal to each other.

By adding or subtracting equally from equal quantities, we obtain equal quantities.

By adding or subtracting from equal values ​​not equally, we obtain unequal values.

By adding or subtracting equally from unequal values, we obtain unequal values.

The sum of the larger ones is greater than the sum of the smaller ones.

A homogeneous quantity, which is no more and no less than another, is equal to it, etc.

Theorem. A theorem or assumption is a truth that requires proof..

Proof is a set of arguments that make the theorem obvious.

The theorem is proved with the help of axioms.

The composition of the theorem. Every theorem consists of a condition and a conclusion.

The condition is sometimes called conjecture, supposition, and the conclusion is sometimes called consequence. The condition is given and therefore sometimes gets the name given.

A theorem is called inverse if the conclusion becomes a condition, and the condition or assumption becomes a conclusion. In this case, this theorem is called a direct one. Not every theorem has its inverse.

Problem or challenge there is a question that can be solved with the help of theorems.

Lemma is an auxiliary truth that facilitates the proof of the theorem.