Presentation on the topic inscribed and described circle. Circle, described near a triangle

What picture is the circumference inscribed in the triangle?

If the circle is inscribed in a triangle,

then the triangle is described near the circumference.

Theorem. In a triangle you can enter a circle, and moreover only one. Its center is the intersection point of the triangle bisector.

Dano: ABC

Prove: There is an OCC. (O; R),

inscribed in a triangle

Evidence:

We carry the triangle bisector: AA 1, BB 1, SS 1.

By property (wonderful triangle point)

bisector intersect at one point - oh,

and this point is equidistant from all sides of the triangle, i.e.

OK \u003d OE \u003d OR, where OK AV, OE Sun, OR AC, means

O is the center of the circle, and AB, Sun, AU - tangents to it.

So, the circle is inscribed in ABC.

Dano: OCP. (O; R) inscribed in ABC,

p \u003d ½ (AV + Sun + AC) - half-version.

Prove S. ABC \u003d p · r

Evidence:

connect the center of the circle with vertices

triangle and conduct radius

circle in touch point.

These radii are

heights of the triangles of AOs, Vos, Soa.

S ABC \u003d S AOB + S Boc + S AOC \u003d ½ AB · R + ½ BC · R + ½ AC · R \u003d

\u003d ½ (AB + BC + AC) · R \u003d ½ P · R.

Task: in an equilateral triangle with a side of 4 cm

the circle is inscribed. Find her radius.

The output of the formula for the radius inscribed in the triangle of the circle

S \u003d p · r \u003d ½ p · r \u003d ½ (a + b + c) · r

2s \u003d (A + B + C) · R

The desired formula for the circle radius,

inscribed in a rectangular triangle

- kartets, C - hypotenuse

Definition: the circle is called inscribed in a quadrilateral, if all sides of the quadrolon concern it.

In what figure the circumference is inscribed in the quadricle:

Theorem: if a circle is inscribed in a quadril,

then the sums of opposite sides

the quadricle is equal ( in anyone described

quadril the sum opposites

sides are equal).

AV + SC \u003d Sun + AK.

Reverse theorem.: if the sums of opposite sides

the convex four-brother is equal,

that in it you can enter the circle.

Task: In the rhombus, the sharp corner of which is 60 0, the circle is inscribed,

the radius of which is 2 cm. Find the perimeter of rhombus.

Task Share

Danar: OCP. (O; R) inscribed in AVSK,

R Avsk \u003d 10

Find: Sun + AK

Given: AVSM is described near OCD. (O; R)

Bc \u003d 6, am \u003d 15,

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Signatures for slides:

Described circle

Definition: The circle is called described near the triangle, if all the vertices of the triangle lie on this circle. In what figure the circle is described near the triangle: 1) 2) 3) 4) 5) if the circle is described near the triangle, then the triangle is entered into a circle.

Theorem. Near the triangle you can describe the circle, and moreover only one. Her center is the point of intersection of the middle perpendicular to the sides of the triangle. And in the given: ABC to prove: There is an OCD. (O; R), described about ABC. Proof: We will conduct middle perpendicular P, K, N to the parties of AV, Sun, AU for the property of the middle perpendicular to the sides of the triangle (wonderful triangle point): they intersect at one point - o, for which OA \u003d OS \u003d OS. That is, all the vertices of the triangle are equidistant of the point o, it means that they lie on the circle with the center of O. So, the circle is described near the ABS triangle. O n p k

Important property: if the circle is described near a rectangular triangle, then its center is the middle of the hypotenuse. O R R c a b r \u003d ½ AB Task: find the radius of the circle described near a rectangular triangle whose karts are 3 cm and 4 cm. The center of the circle described near the stupid triangle lies outside the triangle.

a B C R R \u003d Formulas for the radius described near the triangle circumference Task: find the radius of the circle described near the equilateral triangle, the side of which is 4 cm. Solution: R \u003d R \u003d, answer: cm (cm)

Task: In a circle, the radius of which is 10 cm, inscribed an equifiable triangle. The height conducted to its base is 16 cm. Find the side and the area of \u200b\u200bthe triangle. And in C ON Solution: T. K. The circle is described near an equally chained triangle ABC, the center of the circle lies at the height of the VN. AO \u003d CO \u003d CO \u003d 10 cm, it \u003d vn - at \u003d \u003d 16 - 10 \u003d 6 (cm) of the Aon - rectangular, AO 2 \u003d An 2 + An 2, An 2 \u003d 10 2 - 6 2 \u003d 64, An \u003d 8 cm AVN - rectangular, AV 2 \u003d AN 2 + VN 2 \u003d 8 2 + 16 2 \u003d 64 + 256 \u003d 320, AV \u003d (cm) AC \u003d 2AN \u003d 2 · 8 \u003d 16 (cm), s AVS \u003d ½ AS · Vn \u003d ½ · 16 · 16 \u003d 128 (cm 2) Answer: Av \u003d cm S \u003d 128 cm 2, Find: Av, S ABC Dano: AVS-P / B, VN AS, VN \u003d 16 cm OK. (About ; 10 cm) described about ABC

Definition: The circle is called described about a quadrilateral, if all the vertices of the quadricle lie on the circle. Theorem. If about a quadrilateral is described a circle, then the sum of its opposite angles is 180 0. Proof: T. K. The circle is described about ABC D, then A, B, C, D - inscribed, it means, a + c \u003d ½ BCD + ½ BAD \u003d ½ (BCD + BAD) \u003d ½ · 360 0 \u003d 180 0 b + D \u003d ½ ADC + ½ ABC \u003d ½ (ADC + ABC) \u003d ½ · 360 0 \u003d 180 0 A + C \u003d B + D \u003d 180 0 is given: OCC. (O; R) described about AVS D to prove: it means a + C \u003d b + d \u003d 180 0 Other formulation of the theorem: The sum of the opposite angles is 180 inscribed in the circumference of the quadrope. A b c d

Reverse theorem: If the sum of the opposite angles of the quadrilateer is 180 0, then the circle can be described near it. It is given: ABC D, A + C \u003d 180 0 A B C D about prove: OCC. (O; R) described about ABC D Proof: No. 729 (textbook) around which quadrolon can not be described around the circle?

Corollary 1: Near any rectangle you can describe the circle, its center is the point of intersection of diagonals. Corollary 2: A circle can be described near an equilibrium trapezion. And in with to

Share Tasks 80 0 120 0? ? A V S M K N O R E 70 0 Find the corners of a quadrotter RCEN: 80 0

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Grade 8 L.S. Atanasyan Geometry 7-9 inscribed and described circle

O D C If all sides of the polygon touch the circumference, the circumference is called inscribed in the polygon. A E A polygon is called described near this circle.

D with which of the two quadrangles of ABC D or AEK D is described? And e to about

D in with a rectangle can not enter the circle. And O.

D In what famous properties will be useful to us when studying the inscribed circle? A E about the property of a tangent property of tangent segments f p

D C in any described quadrangle sums of opposite sides are equal. A E O A A R N F B B C C C D D

D C The sum of the two opposite sides of the quadrangle described is 15 cm. Find the perimeter of this quadrangle. And about number 695 in C + AD \u003d 15 AB + DC \u003d 15 P ABCD \u003d 30 cm

D f Find FD A O N? 4 7 6 5

D in with an equilibrium trapezium is described near the circle. The bases of the trapezium are equal to 2 and 8. Find the radius of the inscribed circle. And in c + ad \u003d 1 0 ab + dc \u003d 1 0 2 8 5 5 2 n f 3 3 4 s l

D B is true and reverse statement. And o If the sums of the opposite sides of the convex quadrilateral are equal, then it can be inserted into it. Sun + A D \u003d AV + DC

D in C is it possible to enter a circle in this quadrilateral? A O 5 + 7 \u003d 4 + 8 5 7 4 8

In C and in any triangle, you can enter a circle. Theorem Prove that in a triangle you can enter the circle is given: ABC

K in C and L M o 1) DP: bisector of triangle angles 2) with OL \u003d CO M, on hypotenuse and east. The angle of L \u003d M is carried out from the point of perpendicular to the sides of the triangle 3) of the MOA \u003d CoA, on hypotenuse and the OST. The angle of MO \u003d KO 4) L O \u003d M O \u003d K about the point about equidistant from the side of the triangle. So, the circle with the center in So passes through points k, l and m. The parties of the ABC triangle touches this circle. So the circle is inscribed by ABC.

K in C and in any triangle you can enter a circle. L M About Theorem

D in with prove that the area of \u200b\u200bthe described polygon is equal to half the work of its perimeter on the radius of the inscribed circle. A No. 69 7 F R A 1 A 2 A 3 R O R ... + K

O D C If all the tops of the polygon lie on the circle, the circle is called the described polygon. A E A polygon is called inscribed in this circle.

About D with which of the polygons depicted in the figure is inscribed in a circle? A E L P X E O D B C A E

O and in D with what famous properties will be useful to us when studying the circumference described? Theorem on the inscribed coal

O and in D In any inscribed quadrangle, the sum of the opposite angles is 180 0. C + 360 0

59 0? 90 0? 65 0? 100 0 D A B C O 80 0 115 0 D A B C O 121 0 Find unknown corners of quadrangles.

D is true and reverse statement. If the sum of the opposite angles of the quadrangle is 180 0, then the circle can be written near it. A B C O 80 0 100 0 113 0 67 0 O D A B C 79 0 99 0 123 0 77 0

In C and about any triangle, you can describe the circle. Theorem Prove that the circle can be described: ABC

K in C and lm o 1) DP: middle perpendicular to sides \u003d CO 2) in OL \u003d CO L, according to categories 3) som \u003d and o m, according to CATEMS CO \u003d AO 4) at \u003d co \u003d JSC, t. e. The point is equal to the vertices of the triangle. So, a circle with the center in T.OO and the radius of OA will pass through all three vertices of the triangle, i.e. It is the described circle.

K in C and about any triangle you can describe the circle. L M Theorem O

The triangle of the ABC so that the circle diameter is inscribed in C and C and C C A No. 702. Find the corners of the triangle, if: a) Sun \u003d 134 0 134 0 67 0 23 0 b) ac \u003d 70 0 70 0 55 0 35 0

About in C and No. 703 Incidentally inscribed an equifiable triangle ABC with the base of the aircraft. Find the corners of the triangle if Sun \u003d 102 0. 102 0 51 0 (180 0 - 51 0): 2 \u003d 129 0: 2 \u003d 128 0 60 /: 2 \u003d 64 0 30 /

O in C and No. 704 (a) Circumfied with the center O is described near a rectangular triangle. Prove that the point is the middle of the hypotenuse. 180 0 d and a m e t p

O in C № 704 (b) Circumstial with the center O is described near a rectangular triangle. Find the sides of the triangle if the circle diameter is equal to D, and one of the sharp corners of the triangle is equal. D.

About with C No. 705 (a) Near the rectangular triangle of ABC with a direct angle with a circle described. Find the radius of this circumference if the spear \u003d 8 cm, Sun \u003d 6 cm. 8 6 10 5 5

About with and in № 705 (b) near the rectangular triangle ABC with a direct angle with a circle described. Find the radius of this circle if the speaker \u003d 18 cm, 18 30 0 36 18 18

O in C and the side sides of the triangle shown in the figure are 3 cm. Find the radius of the circumference described near him. 180 0 3 3

O in C and the radius of the circle described near the triangle shown in the drawing is 2 cm. Find the side of AV. 180 0 2 2 45 0?

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