Theorems and general information

I. Geometry

II. Planimetry without formulas.

Two angles are called adjacent If they have one side, and the other two of these angles are additional semicircles.

1. The sum of adjacent angles is equal to 180 ° .

Two angles are called verticalIf the sides of the same corner are additional semicircles of the other.

2. Vertical angles are equal.

Angle equal to 90 ° , called direct angle. Direct, intersecting at right angles are called perpendicular.

3. Through each point, direct can be carried out and moreover, only one perpendicular to the straight line.

Angle, smaller 90 ° , called sharp. Corner greater than 90. ° , called stupid.

4. Signs of the equality of triangles.

- on two sides and the corner between them;

- on the side and two angles adjacent to it;

- for three parties.

Triangle called equaluedIf he has two sides equal.

Median The triangle is called a segment connecting the vertex of the triangle from the middle of the opposite side.

Bissectrice The triangle is called a straight segment concluded between the vertex and the point of its intersection with the opposite side, which divides the angle in half.

Heightthe triangle is a segment of a perpendicular, lowered from the vertex of the triangle on the opposite direction, or to its continuation.

The triangle is called rectangularIf he has a straight corner. In a rectangular triangle, the side opposite to direct corner is called hypotenuse. The remaining two sides are called catetie.

5. Properties of parties and corners of a rectangular triangle:

- cOALS, opposite categories - sharp;

- hypotenuse more than any of the cathets;

- the amount of cathets is more hypotenuse.

6. Signs of equality of rectangular triangles:

- on cathette and acute corner;

- in two categories;

- on hypotenuse and cathetu;

- on hypotenuse and acute corner.

7. Properties of an equal triangle:

- in an equilibried triangle, the angles at the base are equal;

- if in the triangle two corners are equal, then it is an isolated;

In an equilibried triangle, the median conducted to the base is bisector and height;

- if in the triangle of the median and bisector (or height and bisector, or median and height), conducted from any vertex, coincide, then such a triangle is a preceded.

8. In the triangle, against most of the parties, there is a bigger angle, against the larger angle lies the biggest side.

9. (Triangle inequality). Each triangle has the sum of two sides more than a third party.

External angle ABC triangle at vertex A is called an angle, adjacent triangle angle at the top a.

10. The sum of the internal corners of the triangle:

The sum of any two corners of the triangle is less than 180 ° ;

In each triangle, two angles are sharp;

The external angle of the triangle is more than any inner corner, not related to it;

The sum of the corners of the triangle is equal to 180 ° ;

The external angle of the triangle is equal to the sum of the two other corners, not related to it.

The sum of the sharp corners of the rectangular triangle is equal to 90 ° .

The segment connecting the middle sides of the triangle is called the middle line of triangles.

11. The middle line of the triangle has a property - it is parallel to the base of the triangle and is equal to its half.

12. The length of the broken is not less than the length of the segment connecting its ends.

13. Properties of the middle perpendicular segment:

The point lying on the middle perpendicular is equally removed from the segment ends;

Any point, equally removed from the ends of the segment lies on the middle perpendicular.

14. Properties of bisector angle:

Any point lying on the bisector of the angle is equally removed from the side of the angle;

Any point equally removed from the side of the angle lies on the bisector of the angle.

15. The existence of the circle described near the triangle:

All three middle perpendicular triangles intersect at one point and this point is the center of the described circle. The circle described near the triangle always exists and it is the only one;

The center of the described circle of the rectangular triangle is the middle of the hypotenuse.

16. The existence of a circle insistence included in the triangle:

All three triangle bisectors intersect at one point and this point is the center of the inscribed circle. Included in the triangle circle always exists and it is the only one.

17. Signs of direct parallelism. Theorems on parallelism and perpendicularity of direct:

Two straight, parallel to the third - parallel;

If with the intersection of two straight third, internal (outer) will increase the underlying angles, or internal (external) unilateral angles in the amount are equal to 180 ° then these straight parallels;

If parallel straight cross is crossed by the third straight, then the internal and external approximation of the underlying angles are equal, and internal and external one-sided Angles in the amount are equal to 180 ° ;

Two straight, perpendicular to the same straight line - parallel;

Direct, perpendicular to one of two parallel direct, perpendicular and second.

Circle - The set of all points of the plane equidistant from one point.

Chord - Cut connecting two circumference points.

Diameter - Chord, passing through the center.

Tangent - Direct, having one common point with a circle.

Central corner - angle with a vertex in the center of the circle.

Inserted corner - angle with a vertex on the circle, the sides of which crosses the circle.

18. Theorems related to the circle:

The radius carried out to the point of touch is perpendicular to the tangent;

The diameter passing through the middle of the chord is perpendicular to it;

The square of the tangent length is equal to the product of the length of the unit on its outer part;

The central angle is measured by a degree of arc, on which it relies;

The inscribed angle is measured half an arc to which it relies, or complements it half to 180 ° ;

Tangents conducted to the circle from one point are equal;

The product of the sequential on its external part is the value constant;

Parallelogram A quadrangle is called opposite parties parallel.

19. Signs of the parallelogram. Properties Pollogram:

Opposite parties are equal;

Opposite angles are equal;

The diagonal of the parallelogram is divided into the intersection point in half;

The sum of the squares of diagonals is equal to the sum of the squares of all its sides;

If in the convex quadrangle, the opposite parties are equal, then such a quadrilateral - parallelogram;

If in the convex quadrangle, opposite angles are equal, then such a quadrangle - parallelograms;

If the convex quadrangle diagonally is divided into the intersection point in half, then such a quadrangle is parallelograms;

The mid-sides of any quadrilateral are the tops of the parallelogram.

Parallelogram, all sides of which are equal, called rumble.

20. Additional properties and signs of rhombus:

Roma diagonally mutually perpendicular;

Roma diagonals are bisector of its inner corners;

If the diagonal parallelogram is mutually perpendicular, or are bisector of the corresponding angles, then this parallelogram is rhombus.

Parallelogram, all the angles of which are direct, called rectangle.

21. Additional properties and signs of a rectangle:

The diagonal of the rectangle is equal;

If the diagonal parallelogram is equal, then such a parallelogram is a rectangle;

The middle of the sides of the rectangle - the vertices of the rhombus;

The middle of the sides of the rhombus - the vertices of the rectangle.

A rectangle that all parties are equal, called square.

22. Additional properties and signs of the square:

The diagonal of the square is equal and perpendicular;

If the diagonal of the quadrangle is equal and perpendicular, then such a quadrangle is a square.

Quadrilateral, two sides of which are parallel, called trapezium.

The segment connecting the middle of the side of the trapezion is called middle line trapezium.

23. The properties of the trapezium:

- in an equally trapezium, the angles at the base are equal;

- the segment connecting the middle of the diagonals of the trapezium is equal to the duration of the bases of the trapezium.

24. The middle line of the trapezium has a property - it is parallel to the bases of the trapezium and is equal to their half asum.

25. Signs similaritytriangles:

On two corners;

In two proportional parties and the corner between them;

In three proportional parties.

26. Signs of similarity of rectangular triangles:

In acute corner;

On proportional catech;

By proportional cathele and hypotenuse.

27. Relations in polygons:

All the right polygons are similar to each other;

The sum of the angles of any convex polygon is 180 ° (n.-2);

The sum of the external angles of any convex polygon, taken one at each vertex, is 360 ° .

The perimeters of such polygons belong as their similarity Parties, and this ratio is equal to the likeness ratio;

The area of \u200b\u200bsuch polygons belongs as the squares of their similar parties, and this ratio is equal to the square of the likeness coefficient;

The most important planimeters:

28. Falez Theorem. If parallel straight, crossing the side of the angle, cut off on one side of equal segments, then these straight cuts on the other side also equal segments.

29. Pythagora theorem. In a rectangular triangle, the square of the hypotenuse is equal to the sum of the squares of the cathets :.

30. The cosine theorem. In any triangle, the square side is equal to the sum of the squares of the two other sides without their double work on the cosine of the corner between them :.

31. The sinus theorem. The sides of the triangle are proportional to the sinuses of opposite angles: , where - the radius of the circle described near this triangle.

32. Three triangle medians intersect at one point, which divides each median in terms of 2: 1, counting from the top of the triangle.

33. Three straight-containing triangle heights intersect at one point.

34. The parallelogram area is equal to the work of one of its sides to the height, omitted on this side (or the product of the sides of the corner between them).

35. The area of \u200b\u200bthe triangle is equal to half the work side to the height, omitted on this side (or half the work of the parties to the sine corner between them).

36. The area of \u200b\u200bthe trapezium is equal to the work of the base of the bases to height.

37. Roma Square is equal to half the work of diagonals.

38. The area of \u200b\u200bany quadrilateral is equal to half the work of its diagonals on the sine corner between them.

39. Bissektris divides the side of the triangle on segments proportional to two other sides.

40. In a rectangular triangle, a median spent on the hypotenuse, divides the triangle into two isometric triangles.

41. The area of \u200b\u200ban equilibrium trapezium, the diagonal of which is mutually perpendicular, is equal to the square of its height :.

42. The sum of the opposite corners of the quadrangle, inscribed in the circle, is equal to 180 ° .

43. The quadrangle can be described around the circle if the sums of the opposite sides are equal.

III.Basic formulas planimetry.

1. Arbitrary triangle. - from Toron; - opposing corners; - half-meter; - radius of the described circle; - radius inscribed circle; - square; - Height spent on the side:

Solution of rolling triangles:

The cosine theorem :.

Sinus theorem: .

The length of the triangle median is expressed by the formula:

The length of the side of the triangle through the medians is expressed by the formula:

The length of the triangle bisector is expressed by the formula:

Right triangle. - to the athlet; - hypotenuse; - Projections of cathettes on the hypotenuse:

Pythagorean theorem: .

Solution of rectangular triangles:

2. Equilateral triangle:

3. Arbitrary convex quadrangle: - diagonal; - the angle between them; - square.

4. Parallelogram: - Related sides; - the angle between them; - height spent on the side; - square.

5. Rhombus:

6. Rectangle:

7. Square:

8. Trapeze: - grounds; - height or distance between them; - The middle line of the trapezium.

9. Described polygon (- half-meter; - the radius of the inscribed circle):

10. Right polygon (- the side of the right - coal; - radius of the described circle; - Radius inscribed circle):

11. Circle, circle (- Radius; - Circle Length; - Circle Square):

12. Sector (- The length of the arc limiting the sector; - a degree of central angle; - radian measure of the central angle):

Task 1. Area of \u200b\u200ba triangleABC is equal to 30 cm 2. On the sideAC taken point d so that ad: dc \u003d 2: 3. Length perpendicularDE held on the side of BC, equal to 9 cm. FindBC.

Decision. Let's draw bd. (see Fig.1.); TrianglesABD and BDC. have a common heightBF. ; Consequently, their squares refer both to base lengths, i.e.:

AD: DC=2:3,

from 18 cm 2.

On the other hand , or, from where bc \u003d 4 cm. The answer: Bc \u003d 4 cm.

Task 2. In an equilibried height triangle, conducted to the base and to the side, are equal to 10 and 12 cm, respectively. Find the bottom length.

Decision. AT ABC have AB= BC., BD.^ AC, AE^ DC, BD.\u003d 10 cm and AE\u003d 12 cm (see Fig. 2). Let rectangular trianglesAEC. and BDC. Like (corner C. general); Consequently, or 10: 12 \u003d 5: 6. Using Pythagore's theorem to BDC. , we, i.e. .

Transcript.

1 Main definitions, theorems and formulas planimetry. Designations: AVS Triangle with vertices A, B, C. A \u003d BC, B \u003d AS, C \u003d AB of it, respectively, median, bisector, height, conducted to side A, P - perimeter, half-version, R and R radii, respectively Described and inscribed circles. S - Figure area, D 1, D 2 - a quadrangle diagonal, angle between straight a and b; Signs, parallelism, perpendicularity, similarity, respectively. On the definition, t theorem. T 1. (Signs of parallelism of straight lines, Fig. (6). O-1. A 1 in 1 C 1 ", ~ ABC (K is a similarity coefficient) if their parties are proportional to, and the corners are equal to (Fig. 7): Two straight parallel, if: internal passage of underlying angles are equal:< 3 = < 5; внешние накрест лежащие УГЛЫ равны: < 1 = < 7; соответственные углы равны: <1 = < 5; сумма внутренних односторонних углов равна 180: < 2 + < 5= 180 ; сумма внешних односторонних углов равна 180: < 1 + < 6 = 180. Т 2 (признаки подобия). Два треугольника подобны, если: дня угла одного равны двум углам другого; дне стороны одного пропорциональны двум сторонам другого, а углы, заключенные между этими сторонами, равны; три стороны одного пропорциональны трем сторонам другого.

2 T 3. In such triangles, all their linear elements (with the same K) are proportional: sides, medians, bisector, heights, radii inscribed and described circles, etc. T 4 (Falez). Parallel straight, crossing the angles, cut down proportional segments (Fig. 8): T. 5. The sum of the corners of the triangle is 180. t 6. Three medians of the triangle intersect at one point, which divides each median in terms of 2: 1, Considering from the vertex (see Fig. 9): T 7. The middle line of the triangle, connecting the middle of the two sides, parallel to the third side and is equal to its half (Fig. 10): T 8. Bissectrix of the inner corner of the triangle divides the opposite side to the part proportional to the part adjacent parties: BD: CD \u003d AV: AS (see Fig. 11).

3 tons 9. The inscribed angle (formed by two chords emanating from one. The circumference point) is measured half an arc to which it, it relies (Fig. 12): T-10. The central angle formed by two radius of the circumference is measured by the arc on which it relies (see Fig. 12): T 11. The angle between tangential and chord, carried out through the point of touch, is measured by half an arc concluded between its parties (Fig. 13) : T 12. The angle between two sections with a vertex outside the circumference is measured by the semi-of two arcs, prisoners between its parties (Fig. 14): T 13. Tangents conducted to the circumference from a common point located outside the circle are equal to: in A \u003d Sun. The angle between two tangents (described angle) is measured by the high-speed, and smaller arcs concluded between the touch points (Fig. 15):

4 t 14. The angle between two chords with a vertex inside the circle is measured by half a two arcs, one of which is concluded between its parties, the other between their continuations (Fig. 16): T 15. If two chords intersect inside the circles, then the work of segments of one chord Equally, the piece of segments of another (see Fig. 16): OB \u003d with OD. T 16. If a tangential and sequential point was carried out from the point outside the circle, then the tangential square is equal to the product of the segment of the section on its outer part (Fig. 17): T 17. In the rectangular triangle (A, B - kartettes, with hypotenuse. H height, Omitted on the hypotenuse, and C, BC of the projection of cathettes per hypotenuse) take place (Fig. 18): 1. Pythagora formula: C 2 \u003d A 2 + B 2 2. Formula 3. Definition of trigonometric values \u200b\u200b(functions) of sharp corners: 4. Formulas for solving a rectangular triangle:

5 5. The center described near the rectangular triangle of the circle lies in the middle of the hypotenuse and T 18 (sinus theorem). In an arbitrary triangle (Fig. 19) T-19 (cosine theorem). In an arbitrary triangle (Fig. 19): T 20. The sum of the squares of the lengths of the diagonals parallelogram is equal to the sum of the squares of the lengths of its sides: T 21. The center of the circle described in the angle lies on the bisector of this angle. The radius of the circle is perpendicular to the side of the angle and the touch point. The center of the circle inscribed in the triangle is at the point of intersection of the bisector of the triangle angles. T 22. The center of the circle described near the triangle is located at the intersection of the middle perpendicular to the sides. T 23. In the quadrangle described near the circle, the sum of opposite sides are equal. In particular, if an equifiable trapezium is described near the circle, then its middle line is equal to the side. T 24. In the quadrangle inscribed in the circumference, the sum of the opposite angles is equal to 180. T 25. The triangle area is equal to

6 T 26. In the correct triangle with a side A: T 27. In the right N-Oroland (A n, the side of N-square, R radius described, R radius inscribed circumference): T 28. Square of such triangles belongs as the squares of similar parties. O-2. Two figures are called isometric if their square is the same. T 29. Median divides a triangle into two isometric parts. Three medians share a triangle on six isometric parts. Segments connecting the intersection point Median with vertices are divided by a triangle into three isometric parts. T 30. In an arbitrary triangle, the median length is calculated as follows (Fig. 19): T 31. Formulas of the square of quadrangles: Square with a side A: S \u003d A 2; Rectangle with sides n. H Li: S \u003d A B; The parallelogram with the parties a and b: a rhombus with a side A and a sharp angle between the parties: the trapezoids with the bases A and B:

7 convex quadrilateral: T-32. Other formulas: The area of \u200b\u200bthe polygon described near the circle of the radius R: S \u003d P R; Radius Radius Circle Area: System Square (RAD): Length of the Radius Radius R: Arc Length and Or Rad: All Formulas Surface Surface Surface Surface Cuba Side Cube Side Cube Surface Formula Surface Cuba (S):

8 Find the surface area of \u200b\u200bthe rectangular parallelepipeda A, B, C, - the side of the parallelepiped formula of the surface area of \u200b\u200bparallelepipeda, (s): calculation of the surface area of \u200b\u200bthe cylinder R- radius of the cylinder height π 3.14 The side surface of the side surface of the cylinder, (s side): Formula Square of the entire surface of the cylinder, (s): Find the surface area of \u200b\u200bthe ball, formula R - the sphere radius π 3.14

9 Formula of the surface of the ball surface (S): The surface area of \u200b\u200bthe ball sector R is the radius of the ball R - the radius of the base of the cone \u003d radius of the segment π 3.14 The surface area of \u200b\u200bthe ball sector surface, (s): the surface area of \u200b\u200bthe ball layer H is the height of the ball layer, the segment Kn R - The radius of the ball itself O - the center of the ball π 3.14 the formula of the side surface of the ball layer, (s):

10 The surface area of \u200b\u200bthe ball segment The ball segment is part of the ball cut off by the plane. In this example, the ABCD plane. R is the radius of the ball H - the height of the segment segment π 3.14 the surface area of \u200b\u200bthe ball segment surface, (s): the surface area of \u200b\u200bthe correct pyramid through apophem L - apophem (omitted perpendicular OC from the vertex with, on the edge of the AV) P- Perimeter of the base S OUT - The base area of \u200b\u200bthe side surface area of \u200b\u200bthe right pyramid (s side): the formula of the full surface area of \u200b\u200bthe correct pyramid (s):

11 The side surface area of \u200b\u200bthe correct truncated pyramid M - the Pyramid Appeem, the length of the lower base, ABCDE P is the perimeter of the upper base, the abcde of the side surface area of \u200b\u200bthe right truncated pyramid, (s): the surface area of \u200b\u200bthe straight, circular cone R - the radius of the cone base H is the height L - the formula of the cone π 3.14 the formula of the side surface of the cone, through the radius (R) and forming (L), (s side): the formula of the side surface area of \u200b\u200bthe cone, through the radius (R) and height (H), (s side): the formula of the area of \u200b\u200bthe full surface of the cone, through the radius (R) and the forming (L), (s):

12 Formula of the area of \u200b\u200bthe full surface of the cone, through the radius (R) and height (H), (s): the formulas of the surface area of \u200b\u200bthe truncated cone R - the radius of the lower base R- radius of the top base L is the formula for the truncated cone π 3.14 the formula of the side surface of a truncated cone , (S side): the formula of the area of \u200b\u200bthe complete surface of the truncated cone, (s): Calculation of the volume of the cube All formulas for the volume of geometric bodies A is the cube side of the cube formula, (V):

13 The volume of rectangular parallelepipeda A, B, the sides of the parallelepiped formula of the parallelepiped volume, (V): Formula for calculating the volume of the ball R-radius π 3.14 The volume of the ball, (V): The volume of the ball layer H - the height of the ball layer R- radius Lower base R- Upper base radius π 3,14

14 Ball layer volume, (V): The volume of the ball sector H is the height of the segment R - the radius of the ball π 3.14 the volume of the ball sector, (V): the volume of the ball segment, the formula sharovy segment is part of the ball cut off by the plane. In this example, the ABCD plane. R is the radius of the ball H - the height of the segment π 3.14 the volume of the ball segment, (V):

15 How to calculate the volume of the cylinder? H- The height of the cylinder R- radius of the base π 3.14 The volume of the cylinder, (V): How to find a volume of cone? H- The height of the cone R radius π 3.14 The volume of the cone, (V): The formula of the truncated cone R- radius of the lower base R- radius of the top base H- The height of the cone π 3,14

16 Volume of truncated cone, (V): Calculation of the pyramid H - the height of the pyramid S - the base area of \u200b\u200bABCDE Pyramid volume, (V): Calculation of the truncated pyramid H - the height of the pyramid s bottom - the lower base area, ABCDE S is the top - top base area , ABCDE Cured Pyramid, (V): Find the volume of the right pyramid

17 Pyramid at the base, which lies the right polygon and the face of equal triangles, is called proper. H is the height of the pyramid A - the side of the base of the pyramid n - the number of parties of the polygon at the base of the volume of the correct pyramid, (V): the volume of the correct triangular pyramid of the pyramid, in which the base of the equilateral triangle and the edges are equal, equal triangles are called the correct triangular pyramid. H is the height of the pyramid A - the base side of the volume of the correct triangular pyramid, (v): the volume of the correct quadrangular pyramid of the pyramid, in which the base of the square and the edges are equal, equal triangles, are called the correct quadrangular pyramid. H is the height of the pyramid A - the base side volume of the correct quadrangular pyramid, (V):

18 The volume of the correct tetrahedron is the correct tetrahedron-pyramid that all faces, equilateral triangles. A-Breet Tetrahedra Volume of the correct tetrahedron (V):

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The article presents the most important theoretical information and necessary for solving specific objectives of the formula. In the shelves, important statements and properties of figures are decomposed.

Definition and Important Facts

Planimetry is a geometry section that considers objects on a flat two-dimensional surface. You can select some suitable examples: square, circle, rhombus.

Among other things, it is worth highlighting the point and straight. They are two basic concepts of planimetry.

Already on them are all the rest, for example:

Axioms and Theorems

We will understand more about the axioms. In planimetry, these are the most important rules for which all science works. And not only in it. By definition, we are talking about allegations that do not require evidence.

Axioms that will be discussed below are included in the so-called Euclidean geometry.

There are two points. Through them, you can always hold the only straight.
If there is straight, that is, the points that lie on it, and the points that do not lie on it.

This 2 approval is customary to call axioms of accessories, and the following - about:

If there are three points on the straight, then one of them is necessarily located between two others.
The plane shares any straight into two parts. When the segments are lying on one half, then the whole object belongs to it. Otherwise, the source straight and segment have an intersection point.

Axioms of measures:

Each segment has a length different from zero. If the point breaks it into several parts, then their sum will be equal to the total length of the object.
Each angle has a certain degree, which is not zero. If you split it with a beam, the initial angle will be equal to the amount of formed.

Parallelism:

On the plane is straight. Through any point that does not belong to it, you can spend only one straight, parallel to this.

Theorems in planimetry are no longer quite fundamental allegations. Usually they are taken as a fact, but each of them has the proof, built on the basic concepts mentioned above. In addition, there are a lot of them. It will be quite difficult to disassemble everything, but some of them will be present in the presented material.

With the following two it is necessary to familiarize yourself with early:

The sum of adjacent angles is 180 degrees.
Vertical angles have the same magnitude.

These two theorems can be useful in solving geometric tasks associated with N-cockers. They are pretty simple and intuitive. It is worth remembering them.

Triangles

The triangle is a geometric shape consisting of three successively connected segments. Classify them in several signs.

On the sides (the relationships are floating out of the names):

In the corners:

acute;
rectangular;
stupid.

Two corners, regardless of the situation, will always be sharp, and the third is determined by the first part of the word. That is, a rectangular triangle has one of the corners is 90 degrees.

Properties:

The greater the angle, the more the side opposite to him.
The sum of all angles is 180 degrees.
The area can be calculated by the formula: S \u003d ½ ⋅ H ⋅ A, where A is the side, H is the height of it.
You can always enter the circle in the triangle or describe it around it.

About one of the main formulas of planimity says the theorem Pythagora. It works exclusively for a rectangular triangle and sounds like this: the square of the hypotenuse is equal to the sum of the squares of the cathets: AB 2 \u003d AC 2 + BC 2.

Under hypotenuse, they mean the side, the opposite corner of 90 °, and under the customs - adjacent.

Quadrangles

Information on this topic is extremely much. Below is only the most important.

Some varieties:

Parallelograms - the opposite sides are equal and parallel.
Rhombus - parallelograms whose sides have the same length.
Rectangle - parallelogram with four straight corners
Square - simultaneously rhombus and rectangle.
The trapezium - only two opposite sides are parallel.

Properties:

The inner corners are 360 \u200b\u200bdegrees.
The area can always be calculated by the formula: S \u003d √ (P-a) (P-b) (P-C) (P - D), where P is half the perimeter, a, b, c, d - sides of the figure.
If around the quadrilateer, you can describe the circle, then I call it convex, if not, is unscrew.

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Planimetry

Basic information from school geometry

1. Signs of the equality of triangles.
1) If two sides and the angle between them are one triangle, respectively, are equal to two sides and the corner between them of another triangle, then the triangles are equal.
2) If the side and two adjacent angle of one triangle are appropriate to it, respectively, are equal to the other triangles adjacent to it, then the triangles are equal.
3) If the three sides of one triangle are respectively equal to three sides of another triangle, then the triangles are equal.

2. The main properties and signs of an equifiable triangle.
1) angles at the base of an equifiable triangle are equal.
2) a median of an equifiable triangle conducted to the base is a bisector and height.
3) If two corners of the triangle are equal, then it is an isce.
4) If the median is a triangle is its height, then a triangle
isosceles.
5) If the triangle bisector is its height, then the triangle is a preceded.
6) If the median of the triangle is its bisector, then the triangle is a preceded.

3. The geometric location of the points equal to the ends of the segment is straight, perpendicular to this segment and passing through its middle (middle perpendicular to the segment).

4. Signs and properties of parallel straight lines.
1) axiom parallel. Through this point, you can spend no more than one direct parallel.
2) If, with the intersection of two straight thirds, equal inner will be formed with the internal passage of the underlying angles, then direct are parallel.
3) If two straight parallels are one and the same straight, then they are parallel to each other.
4) Two straight, perpendicular to the same direct, parallel.
5) If two parallel straight crossing of the third, then the internal ease of the underlying angles are equal.

5. The theorem on the sum of the corners of the triangle and the consequences of it.
1) The sum of the internal corners of the triangle is 180◦.
2) The outer angle of the triangle is equal to the sum of the two internal corners not related to it.
3) The sum of the internal angles of the convex N-carbon is 180◦ (N-2).
4) The sum of the outer angles of the N-Corner is 360◦.
5) angles with mutually perpendicular sides are equal if they are both sharp or both stupid.

6. If the bisector of the angles B and C of the ABC triangle intersect at point M, then ∠BMC \u003d 90◦ + ∠A / 2.

7. The angle between the bisectors of adjacent angles is 90◦.

8. The bisector of internal unilateral angles with parallel straight lines and secular perpendicular.

9. Signs of equality of rectangular triangles.
1) in two categories.
2) on cathetu and hypotenuse.
3) on hypotenuse and acute corner.
4) on cathette and acute corner.

10. The geometric location of the inner points of the angle, equidistant from its sides, is the bisector of the angle.

11 . The cattat of the rectangular triangle lying against an angle of 30 ° is equal to half the hypotenuse.

12. If the roll of the rectangular triangle is equal to half the hypotenuse, the angle, opposed to this cathelet, is 30◦.

13. Triangle inequality. The sum of the two sides of the triangle is more than a third party.

14. Consequence of the inequality of the triangle. The sum of the links of the broken more segment connecting the beginning of the first link with the end of the latter.

15. A big side is against the larger corner of the triangle.

16. Against the most side of the triangle lies a larger angle.

17. The hypotenuse of the rectangular triangle is more category.

18. If from one point was carried out to a straight perpendicular and inclined, then
1) perpendicular shorter inclined;
2) a big controversy corresponds to a big projection and vice versa

19. Pollogram. The parallelogram is called a quadrangle, the opposite sides of which are parallel parallel.
Properties and signs of a parallelogram.
1) Diagonal breaks parallelograms into two equal triangular-nickname.
2) the opposite sides of the parallelogram are parallelly equal.
3) The opposite angles of the parallelogram are parallelly equal.
4) The diagonal of the parallelogram intersect and divide the intersection point in half.
5) If the opposite sides of the quadrilateral are pairwise equal, then this quadrangle is a parallelogram.
6) If the two opposite sides of the quadrilateral are equal
And parallel, then this quadrilateral is a parallelogram.
7) If the quadrangle's diagonal is shared by the intersection point in half, then this quadrangle is parallelograms.

20. Rectangle. The rectangle is called a straight corner parallelogram.
Properties and signs of a rectangle.
1) The diagonal of the rectangle is equal.
2) If the diagonal parallelogram is equal, then this parallelogram is a rectangle.

21. Romble. Rumble is called a quadrilateral, all sides of which are equal.
Properties and signs of rhombus.
1) diagonally rhombus perpendicular.
2) diagonally rhombus divide his corners in half.
3) If the parallelogram is diagonally perpendicular, then this parallelogram is rhombus.
4) If the diagonal of the parallelogram is divided into its corners in half, then this parallelogram is rhombus.

22. Square. Square is called a rectangle, all sides of which are equal.

23. The geometric area of \u200b\u200bpoints equidistant from this line is two parallel straight.

24. Falez Theorem. If on one side of the angle to postpone equal segments and through their ends to carry out parallel straight lines, crossing the second side of the angle, then equal segments also put on the second side of the angle.

25. The middle line of the triangle. The segment connecting the middle of the two sides of the triangle is called the middle line of the triangle.
The theorem on the middle line of the triangle. The middle line of the triangle is parallel to the side of the triangle and is equal to its half.

26. Property of the middle of the parties of the quadrangle. The mid-sides of any quadrilateral are the tops of the parallelogram.

27. The theorem on the medians of the triangle. The medians of the triangle intersect at one point and divide it in relation to 2: 1, counting from the top.

28. a) If the median of the triangle is equal to half the side to which it is carried out, the triangle is rectangular.
b) The median of a rectangular triangle, conducted from the vertex of a direct angle, is equal to half the hypotenuse.

29. The trapezium. A trapezium is called a quadrangle, which only has two opposite sides (bases) are parallel. The middle line of the trapezium is called a segment connecting the mid-parallel sides (side sides).
Theorem on the middle line of the trapezium. The middle line of the trapezium is parallel to the grounds and is equal to half a half.

30. The segment connecting the middle of the diagonals of the trapezium is equal to the durability of the bases.

31. The trapezium is called anamentary if its side sides are equal.
Properties and signs of an equifiable trapezium.
1) Angles at the basis of an equalway trapezium are equal.
2) the diagonal of an equally trapezium is equal.
3) If the angles at the base of the trapezium are equal, it is equally chained.
4) If the diagonal of the trapezoid is equal, it is an equifiable.
5) The projection of the lateral side of an equifiable trapezium on the basis is equal to the durability of the grounds, and the projection of the diagonal is a half-form basis.

32. Circle. The circle is the geometric location of the points of the plane, remote from this point, called the center of the circle, at the same positive distance.
Properties of circle.
1) Diameter, perpendicular chord, divides it in half.
2) The diameter passing through the middle of the chord, which is not diameter, is perpendicular to this chord.
3) The middle perpendicular to the chord is passed through the center of the circle.
4) Equal chords are removed from the center of the circumference at equal distances.
5) Chords of the circle, remote from the center at equal distances, are equal.
6) The circumference is symmetrical about any diameter.
7) Arc circumference, concluded between parallel chords, are equal.
8) Of the two chords, the larger is less removed from the center.
9) The diameter is the greatest chord of the circle.

33. Wonderful circumference property. The geometric location of points M, of which the cut AB is visible at a right angle (∠amb \u003d 90◦), there is a circle with a diameter of AB without points A and B.

34. The geometric location of points M, of which the segment AB is visible at an acute angle (∠amb< 90◦) есть внешность круга с диаметром AB без точек прямой AB.

35. The geometric point of points M, of which the AB segment is visible under a stupid angle (∠amb\u003e 90◦), there is an interior of a circle with a diameter of AB without a point of ab.

36. Property of middle perpendicular to the sides of the triangle. The middle perpendicular to the sides of the triangle intersect at one point, which is the center of the circle described near the triangle.

37. The line of centers of two intersecting circles is perpendicular to their common chord.

38. The center of the circle described near the rectangular triangle is the middle of the hypotenuse.

39. The theorem on the heights of the triangle. Straight containing the heights of the triangle intersect at one point.

40. Tangent to the circumference. A straight line having a single common point with a circle is called tangent to the circumference.
1) The tangent is perpendicular to the radius spent on the touch point.
2) if straight l.passing through the point on the circle perpendicular to the radius carried out at this point, then direct L. - tangent to the circumference.
3) If the direct passing through the point M relas to the circle at the points A and B, then Ma \u003d MB.
4) The center of the circle inscribed in the angle lies on the bisector of this angle.
5) Theorem on the triangle bisector. The triangle bisectors intersect at one point, which is the center of the circle inscribed in the triangle

41. The radius of the circle inscribed in the rectangular triangle with Cates A, B and the hypotenneus C is equal to (A + B - C) / 2.

42. If m is a touch point with an AC side of the circle inscribed in the ABC triangle, then am \u003d p - Bc, where P is a half-period triangle.

43. The circle concerns the side of the ABC triangle BC and the seques of AB and AC. Then the distance from the vertex A to the point of touching the circle with a straight AB is equal to the half-versioner of the ABC triangle.

44. The circle inscribed in the ABC triangle relates to the parties AB, BC and AC, respectively, at points K, L and M. If ∠bac \u003d α, then ∠klm \u003d 90◦- α / 2.

45. Dases of the radius of R and R (R\u003e R). The distance between their centers is equal a. (a. \u003e R + R). Then the segments of common external and total internal tangents concluded between the touch points are equal, respectively and

46. If you can enter a circle into a quadrilateral, then the sums of its opposite sides are equal.

47. Concerning the circle. It is said that two circles concern if they have a single common point (touch point).
1) Touch the two circles lies on their lines of centers.
2) The circle of radii R and R with the centers O1 and O2 are outerly due to the times and only if R + R \u003d O1O2.
3) circles of radius R and R (R< R) с центрами O1 и O2 касаются внутренним образом тогда и только тогда, когда R − r = O1O2.
4) Circle with O1 and O2 centers are outer in point K. Some direct concerns these circles at various points A and B and intersect with a total tangent passing through the point K, at the point C. Then ∠AKB \u003d 90◦ and ∠O1CO2 \u003d 90◦.

48. Corners associated with a circle.
1) The angular value of the circumference arc is equal to the angular value of the central angle.
2) the inscribed angle is equal to half the corner of the arc, which it rests.
3) The angle between the intersecting chords is equal to the half as the opposite arcs dried by chords.
4) The angle between the two secuch is equal to the durability of the arcs carved by the sections on the circle.
5) The angle between tangential and chord is equal to half of the angular value of the arc concluded between them.

49. Inserted angles, relying on the same arc equal.

50. The geometric location of the points, from which this segment is visible under this angle, there are two arcs of equal circles (without the ends of these arcs).

51. If the quadrilateral can be entered into a circle, then the sum of its opposite angles is equal to 180◦.

52. If the sum of the opposite angles of the quadrangle is 180◦, then the circle can be described near it.

53. If the trapeze can enter the circle, the lateral side of the trapezium is visible from the center of the circumference at a right angle.

54. If M is a point on the cut of AB, and am: bm \u003d a: b, then AM: AB \u003d A: (A + B), BM: AB \u003d B: (A + B).

55. Theorem on proportional segments. Parallel straight, crossing the angles, carve proportional segments on them.

56. Similarity. Signs like triangles.
1) If two sides of one triangle are respectively proportional to the two sides of the other, and the angles concluded between these sides are equal, then the triangles are similar.
2) If two angle of one triangle is respectively two corners of the other, then triangles are similar.
3) If the three sides of one triangle are respectively proportional to the three sides of the other, then the triangles are similar.

57 . The ratio of the corresponding linear elements of such figures is equal to the likeness ratio.

58. Wonderful trapezium property. The point of intersection of the diagonals of the trapezium, the point of intersection of the continuation of the side and the middle of the bases lie on one straight line.

59. Property of triangle bisector. The bisector of the triangle divides his side into segments proportional to two other sides.

60. The product of the base for height for this triangle is constantly.

61. If BM and CN are the height of the ABC triangle (∠a 90◦), then the AMN triangle is similar to the ABC triangle, and the likeness factor is equal to | COS ∠A |.

62. Producers of cuts of chord chords AB and CD circle intersecting at point E are equal, that is | AE | · | EB | \u003d | CE | · | ED |.

63. Theorem on tangential and sequential and consequence of it.
1) If the tangent and sequent circle are carried out from one point, then the product of the entire section on its external part is equal to the square tangent
2) The product of the entire section on its external part for this point and this circle is constantly.

64. Trigonometric ratios in a rectangular triangle.
1) Roots a rectangular triangle is equal to the product of the hypotenuse on the sine of the opposite or on the cosine of the sharp corner adjacent to this cathelet.
2) Roots a rectangular triangle is equal to another cathette, multiplied by the tangent of the opposite or Kotangenes adjacent to this cathelet of acute angle.

65. Pythagora theorem. The square of the hypotenus of the rectangular triangle is equal to the sum of the squares of the cathets.

66. Theorem, reverse theorem Pythagora. If the SPARD side square is equal to the sum of the squares of the other two of its sides, then the triangle is rectangular.

67. Middle proportional in a rectangular triangle. The height of the rectangular triangle, conducted from the vertex of the direct angle, is the average proportional projections of cathettes on the hypotenuse, and each catt is the average proportional hypotenuse and its projection on the hypotenuse.

68. If you can enter a circle in the trapezion, then the circle's radius is the average proportional segments that the touch point divides the side.

69. The segment of the total outer tangent to two concerning circles of radii R and R is equal to the segment of the total internal tangent, concluded between common external. Both of these segments are equal.

70. Metric relations in the triangle.
1) Cosine theorem. The square side of the triangle is equal to the sum of the squares of the two other sides without a double product of these sides on the cosine of the corner between them.
2) Consequence of the cosine theorem. The sum of the squares of the diagonals of the parallelogram is equal to the sum of the squares of all its sides.
3) Formula for the median of the triangle. If M is a triangle median, conducted to the side C, then , where a and b-station sides of the triangle.
4) Sinus theorem. The sides of the triangle are proportional to the sines of opposite angles.
5) Generalized sinus theorem. The ratio of the side of the triangle to the sinus of an opposite angle is equal to the diameter of the circle described near the triangle.

71. Forms of a triangle square.
1) The area of \u200b\u200bthe triangle is equal to half the product of the base to height.
2) The triangle area is equal to half the work of two sides on the sinus of the corner between them.
3) The area of \u200b\u200bthe triangle is equal to the product of its half-version on the radius of the inscribed circle.
4) The area of \u200b\u200bthe triangle is equal to the product of three sides, divided into the still-stated radius of the circle described.
5) Formula Gerona. where - the half-period of the triangle.

72. Elements of the equilateral triangle with the side a.. Let H, S, R, R be height, area, radii of the described and inscribed circumference of the equilateral triangle with a side a.. Then

73. The formulas of the parallelogram.
1) The area of \u200b\u200bthe parallelogram is equal to the product of the base to height.
2) The area of \u200b\u200bthe parallelogram is equal to the product of its neighboring sides on the corner sinus between them.
3) The rectangle area is equal to the work of two neighboring sides.
4) Roma area is equal to half the work of its diagonals.

74. The area of \u200b\u200bthe trapezium is equal to the work of the base of the base to height.

75. The square of the quadrilateral is equal to half the work of its diagonals on the sine corner between them.

76. The ratio of areas of such triangles is equal to the square of the likeness coefficient.

77. If a circle can be entered into a polygon, then its area is equal to the product of a polygon half-dimmer to the radius of this circle.

78. If m is a point on the ABC triangle BC side, then

79. If p and q - points on the sides of AB and AC (or on their continuations) ABC triangle, then

80. The length of the circle of the radius R is equal to 2πr.
81. The area of \u200b\u200bthe radius R radius is equal to πr 2.

Literature: Gordin R.K., "Every Matshchlinker should know"

Tags ,. Look .

All formulas for planimetry for the exam. International School Scientific Bulletin. The most important theorems of planimetry