1. If the angle of one triangle is equal to the angle of another triangle, then the areas of these triangles are related as the product of the sides enclosing equal angles.
2. Ratio of areas of triangles having general heights, is equal to the ratio of the bases corresponding to these heights.
3. Ratio of areas of triangles having general grounds, is equal to the ratio of the heights corresponding to these sides of the triangle.
4. In similar triangles, similar elements are proportional, the radii of inscribed and circumscribed circles, the perimeters of triangles, square roots from squares.
5.The radius of the inscribed circle can be found using the formula:
6. It is convenient to find the radius of the circumscribed circle using the theorem of sines and cosines:
7.Each median divides the triangle into 2 equal triangles.
8. Three medians divide a triangle into 6 equal triangles.
9. The point of intersection of the bisectors divides the bisector in the ratio:
the sum of the sides forming the angle from which the bisector is drawn to the third side.
10.The medians of a triangle and sides are related by the formula:
11. A straight line parallel to a side of a triangle and intersecting two others cuts off a triangle similar to this one from it.
12. If the bisectors of the anglesB and C triangle ABCintersect at point M, then .
13. Angle between bisectors adjacent corners equals 90.
14. If M is the point of tangency with side AC of a circle inscribed in triangle ABC, then Where - semi-perimeter of a triangle.
15. The circle touches side BC of triangle ABC and the extensions of sides AB and AC. Then the distance from vertex A to the point of contact of the circle with line AB is equal to the semi-perimeter of triangle ABC.
16. A circle inscribed in triangle ABC touches sides AB, BC and AC respectively at pointsK, L And M. If, then.
17. Menelaus's theorem. Given triangle ABC. A certain straight line intersects its sides AB, BC and the continuation of side AC at points C 1, A 1, B 1 respectively. Then
18. Ceva's theorem. Let points A 1, B 1 and C 1 belong respectively to sides BC, AC and AB of triangle ABC. AA segments 1, BB 1 and SS 1 intersect at one point if and only if
19. Steiner-Lemus theorem. If two bisectors of a triangle are equal, then it is isosceles.
20. Stewart's theorem. Dot Dis located on side BC of triangle ABC, then .
21.An excircle is a circle tangent to one of its sides and the extensions of the other two.
22.For each triangle, there are three excircles that are located outside the triangle.
23.The center of an excircle is the point of intersection of the bisectors of the external angles of the triangle and the bisector of the internal one, not adjacent to these two external ones.
24. If the circle touches side BC of triangle ABC and the extensions of sides AB and AC. Then the distance from vertex A to the point of contact of the circle with line AB is equal to the semi-perimeter of triangle ABC.
§ 21. Properties geometric shapes on surface
Lecture 53. Properties of geometric figures on the plane
1. Geometric figures on a plane and their properties
2. Angles, parallel and perpendicular lines
3. Parallel and perpendicular lines
A geometric figure is defined as any set of points. A segment, a straight line, a circle, a ball are geometric shapes.
If all the points of a geometric figure belong to one plane, it is called flat. For example, a segment, a rectangle are flat figures. There are figures that are not flat. This is, for example, a cube, a ball, a pyramid.
Since the concept of a geometric figure is defined through the concept of a set, we can say that one figure is included in another (or contained in another), we can consider the union, intersection and difference of figures.
For example, the union of two rays AB and MK is the straight line KB, and their intersection is the segment AM.
There are convex and non-convex figures. A figure is called convex if, together with any two of its points, it also contains a segment connecting them.
The figures F₁ are convex, and the figure F₂ is non-convex.
Convex figures are a plane, a straight line, a ray, a segment, a point, and a circle.
For polygons, another definition is known: a polygon is called convex if it lies on one side of each straight line containing its side. Since the equivalence of this definition and the one given above for a polygon has been proven, we can use both.
Let's consider some concepts studied in the school geometry course, their definitions and properties, accepting them without proof.
Angles
Corner is a geometric figure that consists of a point and two rays emanating from this point. The rays are called the sides of the angle, and their common beginning is its vertex.
An angle is designated in different ways: either its vertex, or its sides, or three points are indicated: the vertex and points on the sides of the angle: A,(k,l), ABC.
The angle is called expanded, if its sides lie on the same straight line.
An angle that is half a straight angle is called direct. An angle less than a right angle is called sharp. An angle greater than a right angle but less than a straight angle is called stupid.
Flat angle- this is a part of the plane limited by two different rays emanating from one point.
There are two plane angles formed by two rays with a common origin. They're called additional.
ABOUT 
The angles considered in planimetry do not exceed the unfolded angle.
The two angles are called adjacent, if they have one side in common, and the other sides of these angles are additional half-lines.
The sum of adjacent angles is 180º. The validity of this property follows from their definition of adjacent angles.
The two angles are called vertical, if the sides of one angle are complementary half-lines of the sides of the other.
Vertical angles are equal.
Parallel and perpendicular lines
Two lines in a plane are called parallel, if they do not intersect
If line a is parallel to line b, then write a║b.
Let's consider some properties of parallel lines, and first of all, the signs of parallelism.
Signs are theorems that establish the presence of any property of an object in a certain situation. In particular, the need to consider the signs of parallelism of lines is caused by the fact that often in practice it is necessary to resolve the issue of the relative position of two lines, but at the same time it is impossible to directly use the definition.
Consider the following signs of parallel lines:
1. Two lines parallel to a third are parallel to each other.
2. If internal crosswise angles are equal or the sum of internal one-sided angles is equal to 180º, then the lines are parallel.
It is a true statement the opposite the second sign of parallelism of lines: if two parallel lines are intersected by a third, then the internal angles lying across each other are equal, and the sum of one-sided angles is 180º.
An important property of parallel lines is revealed in theorem named after the ancient Greek mathematician Thales: if parallel lines intersecting the sides of an angle cut off equal segments on one side, then they cut off equal segments on the other side.
Two straight lines are called perpendicular if they intersect at right angles.
If line a is perpendicular to line b, then write ab.
The basic properties of perpendicular lines are reflected in two theorems:
1. Through each point of a line you can draw a line perpendicular to it, and only one.
2. From any point not lying on a given line, you can drop a perpendicular to this line, and only one.
A perpendicular to a given line is a segment of a line perpendicular to a given line and ending at their point of intersection. The end of this segment is called the base of the perpendicular.
The length of the perpendicular dropped from a given point to a straight line is called distance from a point to a straight line.
Distance between parallel lines is the distance from any point on one line to another.
Lecture 54. Properties of geometric figures on the plane
4. Triangles, quadrangles, polygons. Formulas for the areas of a triangle, rectangle, parallelogram, trapezoid.
5. Circle, circle.
Triangles
A triangle is one of the simplest geometric shapes. But its study gave birth to a whole science - trigonometry, which arose from practical needs in measuring land plots, drawing up maps of the area, designing various mechanisms.
Triangle is a geometric figure that consists of three points that do not lie on the same line and three pairwise segments connecting them.
Any triangle divides the plane into two parts: internal and external. A figure consisting of a triangle and its interior region is also called a triangle (or planar triangle).
In any triangle, the following elements are distinguished: sides, angles, altitudes, bisectors, medians, midlines.
The angle of a triangle ABC at vertex A is the angle formed by half lines AB and AC.
Height of a triangle dropped from a given vertex is called the perpendicular drawn from this vertex to the line containing the opposite side.
Bisector of a triangle is the bisector segment of an angle of a triangle connecting a vertex to a point on the opposite side.
Median of a triangle drawn from a given vertex is called a segment connecting this vertex with the midpoint of the opposite side.
Middle line of a triangle is the segment connecting the midpoints of its two sides.
Triangles are called congruent if their corresponding sides and corresponding angles are equal. In this case, the corresponding angles must lie opposite the corresponding sides.
In practice and in theoretical constructions, signs of equality of triangles are often used, which provide a faster solution to the question of the relationship between them. There are three such signs:
1. If two sides and the angle between them of one triangle are equal, respectively, to two sides and the angle between them of another triangle, then such triangles are congruent.
2. If the side and adjacent angles of one triangle are equal, respectively, to the side and adjacent angles of another triangle, then such triangles are congruent.
3. If three sides of one triangle are equal to three sides of another triangle, then such triangles are congruent.
The triangle is called isosceles, if its two sides are equal. These equal sides are called lateral, and the third side is called the base of the triangle.
Isosceles triangles have a number of properties, for example:
In an isosceles triangle, the median drawn to the base is the bisector and the altitude.
Let us note several properties of triangles.
1. The sum of the angles of a triangle is 180º.
From this property it follows that in any triangle at least two angles are acute.
2. middle line of a triangle connecting the midpoints of two sides is parallel to the third side and equal to half of it.
3. In any triangle, each side is less than the sum of the other two sides.
For a right triangle, the Pythagorean theorem is true: the square of the hypotenuse equal to the sum squares of legs.
Quadrilaterals
Quadrangle is a figure that consists of four points and four consecutive segments connecting them, and no three of these points should lie on the same line, and the segments connecting them should not intersect. These points are called the vertices of the quadrilateral, and the segments connecting them are called its sides.
Any quadrilateral divides the plane into two parts: internal and external. A figure consisting of a quadrilateral and its interior region is also called a quadrilateral (or planar quadrilateral).
The vertices of a quadrilateral are called adjacent if they are the ends of one of its sides. Vertices that are not adjacent are called opposite. The segments connecting opposite vertices of a quadrilateral are called diagonals.
The sides of a quadrilateral emanating from the same vertex are called adjacent. Sides that do not have a common end are called opposite. In a quadrilateral ABCD, vertices A and B are opposite, sides AB and BC are adjacent, BC and AD are opposite; segments AC and BD are the diagonals of a given quadrilateral.
Quadrilaterals can be convex or non-convex. Thus, the quadrilateral ABCD is convex, and the quadrilateral KRMT is non-convex. Among convex quadrangles, parallelograms and trapezoids are distinguished.
A parallelogram is a quadrilateral whose opposite sides are parallel.
Let ABCD be a parallelogram. From vertex B to straight line AD we draw a perpendicular BE. Then the segment BE is called the height of the parallelogram corresponding to sides BC and AD. Line segment
M 
CM is the height of the parallelogram corresponding to sides CD and AB.
To simplify the recognition of parallelograms, consider the following sign: if the diagonals of a quadrilateral intersect and are divided in half by the intersection point, then this quadrilateral is a parallelogram.
A number of properties of a parallelogram that are not contained in its definition are formulated as theorems and proven. Among them:
1. The diagonals of a parallelogram intersect and are divided in half at the intersection point.
2. A parallelogram has opposite sides and opposite angles equal.
Let us now consider the definition of a trapezoid and its main property.
Trapeze is a quadrilateral whose only two opposite sides are parallel.
These parallel sides are called the bases of the trapezoid. The other two sides are called lateral.
The segment connecting the midpoints of the sides is called the midline of the trapezoid.
The midline of a trapezoid has the following property: it is parallel to the bases and equal to their half-sum.
Of the many parallelograms, rectangles and rhombuses are distinguished.
Rectangle is called a parallelogram in which all angles are right.
Based on this definition, it can be proven that the diagonals of a rectangle are equal.
Diamond is called a parallelogram in which all sides are equal.
Using this definition, we can prove that the diagonals of a rhombus intersect at right angles and are bisectors of its angles.
Squares are selected from many rectangles.
A square is a rectangle whose sides are all equal.
Since the sides of a square are equal, it is also a rhombus. Therefore, a square has the properties of a rectangle and a rhombus.
Polygons
A generalization of the concept of triangle and quadrilateral is the concept of polygon. It is defined through the concept of a broken line.
A broken line A₁A₂A₃…An is a figure that consists of points A₁, A₂, A₃, …, An and the segments A₁A₂, A₂A₃, …, An-₁An connecting them. Points А₁, А₂, А₃, …, Аn are called the vertices of the broken line, and the segments А₁А₂, А₂А₃, …, Аn-₁Аn are its links.
If a broken line has no self-intersections, then it is called simple. If its ends coincide, then it is called closed. About the broken lines shown in the figure we can say: a) – simple; b) – simple closed; c) is a closed broken line that is not simple.
a B C)
The length of a broken line is the sum of the lengths of its links.
It is known that the length of a broken line is not less than the length of the segment connecting its ends.
Polygon A simple closed broken line is called if its neighboring links do not lie on the same straight line.
The vertices of the broken line are called the vertices of the polygon, and its links are called its sides. Line segments connecting non-adjacent vertices are called diagonals.
Any polygon divides the plane into two parts, one of which is called the inner and the other - the outer region of the polygon (or planar polygon).
There are convex and non-convex polygons.
A convex polygon is called regular if all its sides and all angles are equal.
A regular triangle is an equilateral triangle, a regular quadrilateral is a square.
The angle of a convex polygon at a given vertex is the angle formed by its sides converging at this vertex.
It is known that the sum of the angles of a convex n-gon is 180º (n– 2).
In geometry, in addition to convex and non-convex polygons, polygonal figures are also considered.
A polygonal figure is the union of a finite set of polygons.
a B C)
The polygons that make up a polygonal figure may not have common interior points, but they may also have common interior points.
A polygonal figure F is said to consist of polygonal figures if it is their union, and the figures themselves do not have common interior points. For example, the polygonal figures shown in figures a) and c) can be said to consist of two polygonal figures, or that they are divided into two polygonal figures.
Circle and circle
Circumference is a figure that consists of all points of the plane equidistant from a given point, called center.
Any segment connecting a point on a circle to its center is called the radius of the circle. Radius also called the distance from any point on a circle to its center.
A line segment connecting two points on a circle is called chord. The chord passing through the center is called diameter.
A circle is a figure that consists of all points of the plane located at a distance not greater than a given one from a given point. This point is called the center of the circle, and this distance is called the radius of the circle.
The boundary of a circle is a circle with the same center and radius.
Let us recall some properties of the circle and circle.
A line and a circle are said to touch if they have a unique common point. Such a line is called a tangent, and the common point of the line and the circle is called a point of tangency. It has been proven that if a straight line touches a circle, then it is perpendicular to the radius drawn to the point of contact. The converse statement is also true (Fig. a).
A central angle in a circle is a plane angle with a vertex at its center. The part of the circle located inside the plane angle is called the arc of the circle corresponding to this central angle (Fig.b).
An angle whose vertex lies on a circle and whose sides intersect it is called inscribed in this circle (Fig. c).
An angle inscribed in a circle has the following property: it is equal to half the corresponding central angle. In particular, the angles based on the diameter are right angles.
A circle is called circumscribed about a triangle if it passes through all its vertices.
To describe a circle around a triangle, you need to find its center. The rule for finding it is justified by the following theorem:
The center of a circle circumscribed about a triangle is the point of intersection of perpendiculars to its sides drawn through the midpoints of these sides (Fig.a).
A circle is said to be inscribed in a triangle if it touches all its sides.
The rule for finding the center of such a circle is justified by the theorem:
The center of a circle inscribed in a triangle is the intersection point of its bisectors (Fig.b)
Thus, perpendicular bisectors and bisectors intersect at one point, respectively. In geometry it is proven that the medians of a triangle intersect at one point. This point is called the center of gravity of the triangle, and the point of intersection of the altitudes is called the orthocenter.
Thus, in any triangle there are four wonderful points: center of gravity, centers of inscribed and circumscribed circles and orthocenter.
A circle can be circumscribed around any regular polygon, and a circle can be inscribed into any regular polygon, and the centers of the circumscribed and inscribed circles coincide.
Fifth postulate. Discovery of geometries other than Euclid.
To construct geometry, it is enough to select only a few provisions, taking them directly from practice, and using logical reasoning to obtain the rest of the necessary reasoning. The propositions should be called axioms, the consequences from them theorems. The ancient Greek geometer Euclid of Alexandria is the author of the work “Principles”, which lists axioms - provisions, there are 5 of them:
- You can draw a straight line through two points.
- The straight line can be continued in both directions.
- You can draw a circle around any point with an arbitrary radius.
- All right angles are equal to each other.
- If two straight lines in a plane at the intersection with a third form one-sided interior angles, the sum of which is less than two right angles, then these lines intersect (Another formulation: in a plane through a point not lying on a given line, one and only one straight line can be drawn parallel to a given line ).
FORMULATION OF POSTULATE V
This is what the fifth postulate says:
If two straight lines a and b form, at the intersection with a third straight line, internal one-sided angles a and b, the sum of the values of which is less than two right angles (i.e. less than 180°; Fig. 1), then these two straight lines necessarily intersect, and precisely with that side of the third straight line along which angles a and b are located (together constituting less than 180°).
The last fifth postulate drew attention to itself Special attention, because it was formulated much more complexly and was not intuitive like the others. The problem of postulate V was first solved by a professor at Kazan University, the brilliant Russian mathematician Nikolai Ivanovich Lobachevsky (1792-1856), who discovered in 1862. the first non-Euclidean geometry, also called “hyperbolic”.
A set of geometry theorems independent of the Euclidean parallelism axiom , the Hungarian mathematician János Bolyai called "absolute" geometry . All the other theorems, that is, those , in the proof of which we directly or indirectly rely on postulate V, is proper Euclidean geometry.
Among the non-Euclidean geometries the following can be distinguished:
- Geometry of Lobachevsky, Gauss, Bolyai. In a plane, through point A, outside line a, you can draw more than 1 straight line parallel to the given one.
- Spherical geometry. Geometry on the surface of a sphere, the basic facts of which were studied in ancient times in connection with the problems of astronomy. The fact is that the Earth's surface is practically a regular sphere, so geometry was needed to ensure the correctness of calculations under conditions of curved surfaces.
- Riemann geometry. Based on spherical geometry. Riemann significantly expanded the list of theorems and axioms. Riemann geometry is one of the three “great geometries” (Euclid, Lobachevsky and Riemann). If Euclidean geometry is realized on surfaces with constant zero Gaussian curvature, Lobachevsky - with constant negative one, then Riemann geometry is realized on surfaces with constant positive Gaussian curvature. In Riemann geometry, a straight line is defined by two points, a plane by three, two planes intersect along a straight line, etc., but through a given point no parallel can be drawn to a straight line. In particular, this geometry has a theorem: the sum of the angles of a triangle is greater than two straight lines. Historically, Riemann geometry appeared later than the other two geometries (in 1854). Riemann geometry is similar to spherical geometry, but differs in that any two “straight lines” have not two, as in spherical, but only one point of intersection. Therefore, sometimes Riemann geometry is called geometry on a sphere in which opposite points are identified; Thus, a projective plane is obtained from the sphere.
The special role of postulate V, its greater complexity and less clarity (compared to other axioms) led to the fact that mathematicians of later centuries began to try to prove this postulate as a theorem. Some of them tried to derive this postulate from the remaining axioms of Euclid, without adding new statements to them; others openly replaced postulate V with another axiom, which they considered simpler and more obvious. Of course, the new axiom contained a statement equivalent to postulate V. But the analysis of those proofs in which the V postulate was not openly replaced by another axiom shows that statements equivalent to the V postulate were also used here, but this was done implicitly, unnoticed by the author of the proof.
The importance of the fifth postulate cannot be overestimated, since none of the two geometries known to us would exist without it. If scientists had not considered the fifth postulate, then this would not have happened. greatest discovery, because with the help of non-Euclidean geometry, people gained a new understanding of space. It was with the fifth postulate that it all began: it is the starting point, the engine of science.
Intuition suggested that both Euclidean and non-Euclidean geometry are examples of full-fledged mathematics.
Definition and properties of basic geometric shapes.
Basic properties
1. If the angle of one triangle is equal to the angle of another triangle, then the areas of these triangles are related as the product of the sides enclosing equal angles.
2. The ratio of the areas of triangles that have common heights is equal to the ratio of the bases corresponding to these heights.
3. The ratio of the areas of triangles that have common bases is equal to the ratio of the heights corresponding to these sides of the triangle.
4. In similar triangles, similar elements, radii of inscribed and circumscribed circles, perimeters of triangles, square roots of areas are proportional.
5.The radius of the inscribed circle can be found using the formula:
6. It is convenient to find the radius of the circumscribed circle using the theorem of sines and cosines:
7.Each median divides the triangle into 2 equal triangles.
8. Three medians divide a triangle into 6 equal triangles.
9. The point of intersection of the bisectors divides the bisector in the ratio:
the sum of the sides forming the angle from which the bisector is drawn to the third side.
10.The medians of a triangle and sides are related by the formula:
11. A straight line parallel to a side of a triangle and intersecting two others cuts off a triangle similar to this one from it.
12. If the bisectors of angles B and C triangle ABC intersect at point M, then .
13. The angle between the bisectors of adjacent angles is 90.
14. If M is the point of tangency with side AC of a circle inscribed in triangle ABC, then where is the semi-perimeter of the triangle.
15. The circle touches side BC of triangle ABC and the extensions of sides AB and AC. Then the distance from vertex A to the point of contact of the circle with line AB is equal to the semi-perimeter of triangle ABC.
16. A circle inscribed in triangle ABC touches sides AB, BC and AC, respectively, at points K, L and M. If , then .
17.Menelaus's theorem. Given triangle ABC. A certain straight line intersects its sides AB, BC and the continuation of side AC at points C1, A1, B1, respectively. Then
18.Ceva's theorem. Let points A1, B1 and C1 belong respectively to sides BC, AC and AB of triangle ABC. Segments AA1, BB1 and CC1 intersect at one point if and only if
19.Steiner-Lemus theorem. If two bisectors of a triangle are equal, then it is isosceles.
20.Stewart's theorem. Point D is located on side BC of triangle ABC, then .
21.An excircle is a circle tangent to one of its sides and the extensions of the other two.
22.For each triangle, there are three excircles that are located outside the triangle.
23.The center of an excircle is the point of intersection of the bisectors of the external angles of the triangle and the bisector of the internal one, not adjacent to these two external ones.
24. If the circle touches side BC of triangle ABC and the extensions of sides AB and AC. Then the distance from vertex A to the point of contact of the circle with line AB is equal to the semi-perimeter of triangle ABC.
Construction tasks.
Planimetry is a branch of geometry in which figures on a plane are studied.
Figures studied by planimetry:
3. Parallelogram (special cases: square, rectangle, rhombus)
4. Trapezoid
5. Circumference
6. Triangle
7. Polygon
1) Point:
In geometry, topology and related branches of mathematics, a point is an abstract object in space that has neither volume, nor area, nor length, nor any other similar characteristics large dimensions. Thus, a point is a zero-dimensional object. A point is one of the fundamental concepts in mathematics.
Point in Euclidean geometry:
A point is one of the fundamental concepts of geometry, so “point” has no definition. Euclid defined a point as something that cannot be divided.
A straight line is one of the basic concepts of geometry.
Geometric straight line (straight line) - an extended, non-curved geometric object that is not closed on both sides, cross section which tends to zero, and the longitudinal projection onto the plane gives a point.
In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly determined by the axioms of geometry.
If the basis for constructing geometry is the concept of distance between two points in space, then a straight line can be defined as a line along which the path is equal to the distance between two points.
3) Parallelogram:
A parallelogram is a quadrilateral whose opposite sides are parallel in pairs, that is, they lie on parallel lines. Special cases of a parallelogram are rectangle, square and rhombus.
Special cases:
Square- a regular quadrilateral or rhombus, in which all angles are right, or a parallelogram, in which all sides and angles are equal.
A square can be defined as: a rectangle whose two adjacent sides are equal;
a rhombus in which all angles are right (any square is a rhombus, but not every rhombus is a square).
Rectangle is a parallelogram in which all angles are right angles (equal to 90 degrees).
Rhombus is a parallelogram in which all sides are equal. A rhombus with right angles is called a square.
4) Trapezoid:
Trapezoid- a quadrilateral with exactly one pair of opposite sides parallel.
1. A trapezoid whose sides are not equal,
called versatile .
2. A trapezoid whose sides are equal is called isosceles.
3. A trapezoid in which one side makes a right angle with the bases is called rectangular .
The segment connecting the midpoints of the lateral sides of a trapezoid is called midline trapezius (MN). The midline of the trapezoid is parallel to the bases and equal to their half-sum.
A trapezoid can be called a truncated triangle, therefore the names of trapezoids are similar to the names of triangles (triangles can be scalene, isosceles, or right-angled).
5) Circumference:
Circle- the geometric locus of points of the plane equidistant from a given point, called the center, at a given non-zero distance, called its radius.
6) Triangle:
Triangle- the simplest polygon having 3 vertices (angles) and 3 sides; part of the plane bounded by three points and three segments connecting these points in pairs.
7) Polygon:
Polygon- this is a geometric figure, defined as a closed broken line. There are three various options definitions:
Flat closed broken lines;
Plane closed polylines without self-intersections;
Parts of the plane bounded by broken lines.
The vertices of the polygon are called the vertices of the polygon, and the segments are called the sides of the polygon.
Basic properties of a line and a point:
1. Whatever the line, there are points that belong to this line and do not belong to it.
Through any two points you can draw a straight line, and only one.
2. Of the three points on a line, one and only one lies between the other two.
3. Each segment has a certain length greater than zero. The length of a segment is equal to the sum of the lengths of the parts into which it is divided by any of its points.
6. On any half-line from its starting point, you can plot a segment of a given length, and only one.
7. From any half-line, into a given half-plane, you can put an angle with a given degree measure less than 180°, and only one.
8. Whatever the triangle, there is an equal triangle in a given location relative to a given half-line.
Properties of a triangle:
Relationships between sides and angles of a triangle:
1) Opposite the larger side lies the larger angle.
2) The larger side lies opposite the larger angle.
3) Equal angles lie opposite equal sides, and vice versa equal angles lie equal sides.
The relationship between the internal and external angles of a triangle:
1) The sum of any two internal corners triangle is equal outer corner triangle adjacent to the third angle.
2) The sides and angles of a triangle are also related to each other by relations called the theorem of sines and the theorem of cosines.
The triangle is called obtuse, rectangular or acute-angled , if its largest internal angle is respectively greater than, equal to, or less than 90∘.
Middle line of a triangle is the segment connecting the midpoints of two sides of the triangle.
Properties of the midline of a triangle:
1) The line containing the middle line of the triangle is parallel to the line containing the third side of the triangle.
2) The middle line of the triangle is equal to half of the third side.
3) The midline of a triangle cuts off a similar triangle from a triangle.
Rectangle properties:
1) opposite sides are equal and parallel to each other;
2) the diagonals are equal and bisect at the point of intersection;
3) the sum of the squares of the diagonals is equal to the sum of the squares of all (four) sides;
4) rectangles of the same size can completely cover a plane;
5) a rectangle can be divided into two equal rectangles in two ways;
6) the rectangle can be divided into two equal right triangles;
7) around a rectangle you can describe a circle whose diameter is equal to the diagonal of the rectangle;
8) it is impossible to inscribe a circle in a rectangle (except a square) so that it touches all its sides.
Properties of a parallelogram:
1) The middle of the diagonal of a parallelogram is its center of symmetry.
2) Opposite sides of a parallelogram are equal.
3) Opposite angles of a parallelogram are equal.
4) Each diagonal of a parallelogram divides it into two equal triangles.
5) The diagonals of a parallelogram are bisected by the point of intersection.
6) The sum of the squares of the diagonals of a parallelogram (d1 and d2) is equal to the sum of the squares of all its sides: d21+d22=2(a2+b2)
WITH properties of the square:
1) All angles of a square are right, all sides of a square are equal.
2) The diagonals of a square are equal and intersect at right angles.
3) The diagonals of a square divide its angles in half.
Properties of a rhombus:
1. The diagonal of a rhombus divides it into two equal triangles.
2. The diagonals of a rhombus are divided in half at the point of their intersection.
3. The opposite sides of a rhombus are equal to each other, and its opposite angles are equal.
In addition, a rhombus has the following properties:
a) the diagonals of a rhombus are mutually perpendicular;
b) the diagonal of a rhombus divides its angle in half.
Properties of a circle:
1) A straight line may not have common points with a circle; have one common point with the circle (tangent); have two common points with it (secant).
2) Through three points that do not lie on the same line, you can draw a circle, and only one.
3) The point of contact of two circles lies on the line connecting their centers.
Polygon properties:
1) The sum of the interior angles of a flat convex n-gon is equal.
2) The number of diagonals of any n-gon is equal.
3).The product of the sides of a polygon and the sine of the angle between them is equal to the area of the polygon.
4. Triangles, quadrangles, polygons. Formulas for the areas of a triangle, rectangle, parallelogram, trapezoid.
5. Circle, circle.
- Triangles
A triangle is one of the simplest geometric shapes. But its study gave rise to a whole science - trigonometry, which arose from practical needs in measuring land plots, drawing up maps of the area, and designing various mechanisms.
Triangle is a geometric figure that consists of three points that do not lie on the same line and three pairwise segments connecting them.
Any triangle divides the plane into two parts: internal and external. A figure consisting of a triangle and its interior region is also called a triangle (or planar triangle).
In any triangle, the following elements are distinguished: sides, angles, altitudes, bisectors, medians, midlines.
The angle of a triangle ABC at vertex A is the angle formed by half lines AB and AC.
Height of a triangle dropped from a given vertex is called the perpendicular drawn from this vertex to the line containing the opposite side.
Bisector of a triangle is the bisector segment of an angle of a triangle connecting a vertex to a point on the opposite side.
Median of a triangle drawn from a given vertex is called a segment connecting this vertex with the midpoint of the opposite side.
Middle line of a triangle is the segment connecting the midpoints of its two sides.
Triangles are called congruent if their corresponding sides and corresponding angles are equal. In this case, the corresponding angles must lie opposite the corresponding sides.
In practice and in theoretical constructions, signs of equality of triangles are often used, which provide a faster solution to the question of the relationship between them. There are three such signs:
1. If two sides and the angle between them of one triangle are equal, respectively, to two sides and the angle between them of another triangle, then such triangles are congruent.
2. If the side and adjacent angles of one triangle are equal, respectively, to the side and adjacent angles of another triangle, then such triangles are congruent.
3. If three sides of one triangle are equal to three sides of another triangle, then such triangles are congruent.
The triangle is called isosceles, if its two sides are equal. These equal sides are called lateral, and the third side is called the base of the triangle.
Isosceles triangles have a number of properties, for example:
In an isosceles triangle, the median drawn to the base is the bisector and the altitude.
Let us note several properties of triangles.
1. The sum of the angles of a triangle is 180º.
From this property it follows that in any triangle at least two angles are acute.
2. The middle line of the triangle connecting the midpoints of the two sides is parallel to the third side and equal to its half.
3. In any triangle, each side is less than the sum of the other two sides.
For a right triangle, the Pythagorean theorem is true: the square of the hypotenuse is equal to the sum of the squares of the legs.
- Quadrilaterals
Quadrangle is a figure that consists of four points and four consecutive segments connecting them, and no three of these points should lie on the same line, and the segments connecting them should not intersect. These points are called the vertices of the quadrilateral, and the segments connecting them are called its sides.
Any quadrilateral divides the plane into two parts: internal and external. A figure consisting of a quadrilateral and its interior region is also called a quadrilateral (or planar quadrilateral).
The vertices of a quadrilateral are called adjacent if they are the ends of one of its sides. Vertices that are not adjacent are called opposite. The segments connecting opposite vertices of a quadrilateral are called diagonals.
The sides of a quadrilateral emanating from the same vertex are called adjacent. Sides that do not have a common end are called opposite. In a quadrilateral ABCD, vertices A and B are opposite, sides AB and BC are adjacent, BC and AD are opposite; segments AC and ВD are the diagonals of this quadrilateral.
Quadrilaterals can be convex or non-convex. Thus, the quadrilateral ABCD is convex, and the quadrilateral KRMT is non-convex. Among convex quadrangles, parallelograms and trapezoids are distinguished.
A parallelogram is a quadrilateral whose opposite sides are parallel.
Let ABCD be a parallelogram. From vertex B we drop a perpendicular BE to line AD. Then the segment BE is called the height of the parallelogram corresponding to sides BC and AD. Line segment
M 
CM is the height of the parallelogram corresponding to sides CD and AB.
To simplify the recognition of parallelograms, consider the following sign: if the diagonals of a quadrilateral intersect and are divided in half by the intersection point, then this quadrilateral is a parallelogram.
A number of properties of a parallelogram that are not contained in its definition are formulated as theorems and proven. Among them:
1. The diagonals of a parallelogram intersect and are divided in half at the intersection point.
2. A parallelogram has opposite sides and opposite angles equal.
Let us now consider the definition of a trapezoid and its main property.
Trapeze is a quadrilateral whose only two opposite sides are parallel.
These parallel sides are called the bases of the trapezoid. The other two sides are called lateral.
The segment connecting the midpoints of the sides is called the midline of the trapezoid.
The midline of a trapezoid has the following property: it is parallel to the bases and equal to their half-sum.
Of the many parallelograms, rectangles and rhombuses are distinguished.
Rectangle is called a parallelogram in which all angles are right.
Based on this definition, it can be proven that the diagonals of a rectangle are equal.
Diamond is called a parallelogram in which all sides are equal.
Using this definition, we can prove that the diagonals of a rhombus intersect at right angles and are bisectors of its angles.
Squares are selected from many rectangles.
A square is a rectangle whose sides are all equal.
Since the sides of a square are equal, it is also a rhombus. Therefore, a square has the properties of a rectangle and a rhombus.
- Polygons
A generalization of the concept of triangle and quadrilateral is the concept of polygon. It is defined through the concept of a broken line.
A broken line A₁A₂A₃...An is a figure that consists of points A₁, A₂, A₃, ..., An and the segments A₁A₂, A₂A₃, ..., An-₁An connecting them. Points А₁, А₂, А₃, …, Аn are called the vertices of the broken line, and the segments А₁А₂, А₂А₃, …, Аn-₁Аn are its links.
If a broken line has no self-intersections, then it is called simple. If its ends coincide, then it is called closed. About the broken lines shown in the figure we can say: a) – simple; b) – simple closed; c) is a closed broken line that is not simple.
A B C)
The length of a broken line is the sum of the lengths of its links.
It is known that the length of a broken line is not less than the length of the segment connecting its ends.
Polygon A simple closed broken line is called if its neighboring links do not lie on the same straight line.
The vertices of the broken line are called the vertices of the polygon, and its links are called its sides. Line segments connecting non-adjacent vertices are called diagonals.
Any polygon divides the plane into two parts, one of which is called the inner and the other - the outer region of the polygon (or planar polygon).
There are convex and non-convex polygons.
A convex polygon is called regular if all its sides and all angles are equal.
A regular triangle is an equilateral triangle, a regular quadrilateral is a square.
The angle of a convex polygon at a given vertex is the angle formed by its sides converging at this vertex.
It is known that the sum of the angles of a convex n-gon is 180º (n – 2).
In geometry, in addition to convex and non-convex polygons, polygonal figures are also considered.
A polygonal figure is the union of a finite set of polygons.
A B C)
The polygons that make up a polygonal figure may not have common interior points, but they may also have common interior points.
A polygonal figure F is said to consist of polygonal figures if it is their union and the figures themselves do not have common interior points. For example, the polygonal figures shown in figures a) and c) can be said to consist of two polygonal figures, or that they are divided into two polygonal figures.
- Circle and Circle
Circumference is a figure that consists of all points of the plane equidistant from a given point, called center.
Any segment connecting a point on a circle to its center is called the radius of the circle. Radius also called the distance from any point on a circle to its center.
A line segment connecting two points on a circle is called chord. The chord passing through the center is called diameter.
A circle is a figure that consists of all points of the plane located at a distance not greater than a given one from a given point. This point is called the center of the circle, and this distance is called the radius of the circle.
The boundary of a circle is a circle with the same center and radius.
Let us recall some properties of the circle and circle.
A line and a circle are said to touch if they have a single point in common. Such a line is called a tangent, and the common point of the line and the circle is called a point of tangency. It has been proven that if a straight line touches a circle, then it is perpendicular to the radius drawn to the point of contact. The converse statement is also true (Fig. a).
A central angle in a circle is a plane angle with a vertex at its center. The part of the circle located inside the plane angle is called the arc of the circle corresponding to this central angle (Fig.b).
An angle whose vertex lies on a circle and whose sides intersect it is called inscribed in this circle (Fig. c).
An angle inscribed in a circle has the following property: it is equal to half the corresponding central angle. In particular, the angles based on the diameter are right angles.
A circle is called circumscribed about a triangle if it passes through all its vertices.
To describe a circle around a triangle, you need to find its center. The rule for finding it is justified by the following theorem:
The center of a circle circumscribed about a triangle is the point of intersection of perpendiculars to its sides drawn through the midpoints of these sides (Fig.a).
A circle is said to be inscribed in a triangle if it touches all its sides.
The rule for finding the center of such a circle is justified by the theorem:
The center of a circle inscribed in a triangle is the intersection point of its bisectors (Fig.b)
Thus, the perpendicular bisectors and bisectors intersect at one point, respectively. In geometry it is proven that the medians of a triangle intersect at one point. This point is called the center of gravity of the triangle, and the point of intersection of the altitudes is called the orthocenter.
Thus, in any triangle there are four remarkable points: the center of gravity, the centers of the inscribed and circumscribed circles, and the orthocenter.
A circle can be circumscribed around any regular polygon, and a circle can be inscribed into any regular polygon, and the centers of the circumscribed and inscribed circles coincide.
