Download presentation inscribed and circumscribed circle. Circle circumscribing a triangle
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In which figure is a circle inscribed in a triangle?
If a circle is inscribed in a triangle,
then the triangle is circumscribed about the circle.
Theorem. A circle can be inscribed in a triangle, and moreover, only one. Its center is the intersection point of the bisectors of the triangle.
Given: ABC
Prove: there exists Osp.(O; r),
inscribed in a triangle
Proof:
Let's draw the bisectors of the triangle: AA 1, BB 1, SS 1.
By property (remarkable point of the triangle)
bisectors intersect at one point - O,
and this point is equidistant from all sides of the triangle, i.e.:
OK \u003d OE \u003d OR, where OK AB, OE BC, OR AC, then
O is the center of the circle, and AB, BC, AC are tangents to it.
So the circle is inscribed in ABC.
Given: Okr. (O; r) is inscribed in ABC,
p \u003d ½ (AB + BC + AC) - half-perimeter.
Prove: S ABC = p r
Proof:
connect the center of the circle with the vertices
triangle and draw the radii
circles at points of contact.
These radii are
heights 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: into an equilateral triangle with a side of 4 cm
inscribed circle. 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 desired formula for the radius of a circle,
inscribed in right triangle
- legs, c - hypotenuse
Definition: A circle is said to be 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.
Inverse theorem: if the sums of opposite sides
convex quadrilateral are equal,
then a circle can be inscribed in it.
Task: in a rhombus, sharp corner which 60 0 , a circle is inscribed,
whose radius is 2 cm. Find the perimeter of the rhombus.
Solve problems
Given: Okr. (O; r) is inscribed in ABSK,
P ABSC = 10
Find: BC + AK
Given: ABSM is described around approx. (O; r)
BC=6, AM=15,
slide 1
slide 2
Definition: A circle is said to be circumscribed about a triangle if all the vertices of the triangle lie on this circle. If a circle is circumscribed about a triangle, then the triangle is inscribed in the circle.
slide 3
Theorem. A circle can be circumscribed about a triangle, and moreover, only one. Its center is the point of intersection of the midperpendiculars to the sides of the triangle. Proof: Let's draw the perpendicular bisectors p, k, n to the sides AB, BC, AC By the property of the perpendicular bisectors to the sides of the triangle (a wonderful point of the triangle): they intersect at one point - O, for which OA \u003d OB \u003d OS. That is, all the vertices of the triangle are equidistant from the point O, which means that they lie on a circle with center O. This means that the circle is circumscribed near the triangle ABC.
slide 4
Important property: If a circle is circumscribed near a right triangle, then its center is the midpoint of the hypotenuse. R \u003d ½ AB Task: find the radius of a circle circumscribed about a right triangle whose legs are 3 cm and 4 cm.
slide 5
Formulas for the radius of a circle circumscribed about a triangle Task: find the radius of a circle circumscribed about an equilateral triangle whose side is 4 cm. Solution:
slide 6
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 lateral side and the area of the triangle. Solution: Since the circle is circumscribed about isosceles triangle ABC, then the center of the circle lies at height BH. AO = VO = CO = 10 cm, OH = VN - VO = = 16 - 10 = 6 (cm) AC = 2AH = 2 8 = 16 (cm), SABC = ½ AC WH = ½ 16 16 128 (cm2)
Slide 7
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 near a quadrilateral, then the sum of its opposite angles is equal to 1800. Proof: Another formulation of the theorem: in a quadrangle inscribed in a circle, the sum of opposite angles is equal to 1800.
Slide 8
Converse theorem: if the sum of the opposite angles of a quadrilateral is 1800, then a circle can be circumscribed around it. Proof: № 729 (textbook) Around which quadrilateral is it impossible to circumscribe a circle?
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Properties of a triangle and a trapezoid inscribed in a circle obtuse-angled tr-ka, does not lie in the tr-ke
