A circle circumscribed around a triangle. Presentation on geometry "inscribed and circumscribed circle" then the sums of opposite sides

In which picture is a circle inscribed in a triangle?

If a circle is inscribed in a triangle,

then the triangle is circumscribed about a circle.

Theorem. You can inscribe a circle in a triangle, and only one. Its center is the point of intersection of the bisectors of the triangle.

Given by: ABC

Prove: there is Env.(O; r),

inscribed in a triangle

Proof:

Let's draw the bisectors of the triangle: AA 1, BB 1, СС 1.

By property (remarkable point of the triangle)

bisectors intersect at one point - Oh,

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

OK = OE = OR, where OK AB, OE BC, OR AC, which means

O is the center of the circle, and AB, BC, AC are tangents to it.

This means that the circle is inscribed in ABC.

Given: Environment (O; r) is inscribed in ABC,

p = ½ (AB + BC + AC) – semi-perimeter.

Prove: S ABC = p r

Proof:

connect the center of the circle with the vertices

triangle and draw the radii

circles at the points of contact.

These radii are

altitudes of triangles AOB, BOC, COA.

S ABC = S AOB +S BOC + S AOC = ½ AB r + ½ BC r + ½ AC r =

= ½ (AB + BC + AC) r = ½ p r.

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

circle is inscribed. Find its radius.

Derivation of the formula for the radius of a circle inscribed in a triangle

S = p r = ½ P r = ½ (a + b + c) r

2S = (a + b + c) r

The required formula for the radius of a circle is

inscribed in a right triangle

- legs, c - hypotenuse

Definition: A circle is called inscribed in a quadrilateral if all sides of the quadrilateral touch it.

In which figure is a circle inscribed in a quadrilateral?

Theorem: if a circle is inscribed in a quadrilateral,

then the sums of opposite sides

quadrilaterals are equal ( in any described

quadrilateral sum of opposites

sides are equal).

AB + SK = BC + AK.

Converse theorem: if the sums of opposite sides

convex quadrilateral are equal,

then you can fit a circle into it.

Problem: a circle is inscribed in a rhombus whose acute angle is 60 0,

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

Solve problems

Given: Env.(O; r) is inscribed in ABCC,

R ABCC = 10

Find: BC + AK

Given: ABCM is described about Environ.(O; r)

BC = 6, AM = 15,

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Slide captions:

Circumcircle

Definition: a circle is called circumscribed about a triangle if all the vertices of the triangle lie on this circle. In which figure is a circle described around a triangle: 1) 2) 3) 4) 5) If a circle is described around a triangle, then the triangle is inscribed in the circle.

Theorem. Around a triangle you can describe a circle, and only one. Its center is the point of intersection of the perpendicular bisectors to the sides of the triangle. A B C Given: ABC Prove: there is an Environment (O; r) described near ABC. Proof: Let us draw perpendicular bisectors p, k, n to the sides AB, BC, AC. According to the property of perpendicular bisectors to the sides of a triangle (a remarkable point of a triangle): they intersect at one point - O, for which OA = OB = OC. That is, all the vertices of the triangle are equidistant from point O, which means they lie on a circle with center O. This means that the circle is circumscribed about triangle ABC. O n p k

Important property: If a circle is circumscribed about right triangle, then its center is the middle of the hypotenuse. O R R C A B R = ½ AB Problem: find the radius of a circle circumscribed about a right triangle, the legs of which are 3 cm and 4 cm. The center of a circle circumscribed about an obtuse triangle lies outside the triangle.

a b c R R = Formulas for the radius of a circle circumscribed by a triangle Task: find the radius of a circle circumscribed by an equilateral triangle whose side is 4 cm. Solution: R = R = , Answer: cm (cm)

Problem: an isosceles triangle is inscribed in a circle with a radius of 10 cm. The height drawn to its base is 16 cm. Find the lateral side and area of ​​the triangle. A B C O N Solution: Since the circle is circumscribed about the isosceles triangle ABC, the center of the circle lies at the height BH. AO = VO = CO = 10 cm, OH = VN – VO = = 16 – 10 = 6 (cm) AON – rectangular, AO 2 = AN 2 + AN 2, AN 2 = 10 2 – 6 2 = 64, AN = 8 cm ABN - rectangular, AB 2 = AN 2 + VN 2 = 8 2 + 16 2 = 64 + 256 = 320, AB = (cm) AC = 2AN = 2 8 = 16 (cm), S ABC = ½ AC · ВН = ½ · 16 · 16 = 128 (cm 2) Answer: AB = cm S = 128 cm 2, Find: AB, S ABC Given: ABC-r/b, VN AC, VN = 16 cm Surround (O ; 10 cm) is described near ABC

Definition: a circle is said to be circumscribed about a quadrilateral if all the vertices of the quadrilateral lie on the circle. Theorem. If a circle is circumscribed around a quadrilateral, then the sum of its opposite angles is equal to 180 0. Proof: Since the circle is circumscribed about ABC D, then A, B, C, D are inscribed, which means A + C = ½ BCD + ½ BAD = ½ (BCD + BAD) = ½ 360 0 = 180 0 B+ D = ½ ADC + ½ ABC = ½ (ADC+ ABC) = ½ 360 0 = 180 0 A + C = B + D = 180 0 Given: Environment (O; R) is described around ABC D Prove: So A + C = B + D = 180 0 Another formulation of the theorem: in a quadrilateral inscribed in a circle, the sum of opposite angles is 180 0. A B C D O

Converse theorem: if the sum of the opposite angles of a quadrilateral is 180 0, then a circle can be described around it. Given: ABC D, A + C = 180 0 A B C D O Prove: Surround (O; R) is described around ABC D Proof: No. 729 (textbook) Which quadrilateral cannot be described around a circle?

Corollary 1: around any rectangle you can describe a circle, its center is the point of intersection of the diagonals. Corollary 2: about isosceles trapezoid can describe a circle. A B C K

Solve problems 80 0 120 0 ? ? A B C M K N O R E 70 0 Find the angles of the quadrilateral RKEN: 80 0

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