Mathematically, if each side of a polygon on a two-dimensional plane can be tangent to a circle inside the polygon, then the circle is the inscribed circle of the polygon, and then the polygon is called a circle circumscribed polygon. It is also the largest circle in a polygon. The center of the inscribed circle is called the center of the polygon. A polygon has at most one inscribed circle, which means that for a polygon, its inscribed circle, if it exists, is unique. Not all polygons have inscribed circles. Triangles and regular polygons must have inscribed circles. A quadrilateral with a circumscribed circle is called a circumscribed quadrilateral.

concept

A circle tangent to all sides of a polygon is called the inscribed circle of the polygon.

In particular, the circle tangent to all three sides of a triangle is called the inscribed circle of the triangle, the center of the circle is called the heart of the triangle, and the triangle is called the circumscribed triangle of the circle. The center of a triangle is the intersection of three bisectors of the triangle.

A triangle must have an inscribed circle, but other figures do not. The center of the inscribed circle is located inside the triangle.

nature

In a triangle, the intersection of bisectors of three angles is the center of the inscribed circle, and the vertical segments from the center to each side of the triangle are equal.

Common auxiliary lines: perpendicular to the center of the circle.

Correlation formula

For a general triangle, the formula of inscribed circle radius is as follows: r=sqrt[(p-a)(p-b)(p-c)/p]

In the inscribed circle of a right triangle, there are two simple formulas: 1. The sum of two right angles minus the hypotenuse and then divided by 2, the number is the radius of the inscribed circle.

The product of two right-angled sides divided by the perimeter of a right-angled triangle is the radius of the inscribed circle.

1, r=(a+b-c)/2 (Note: S is the area of Rt△, A and B are the two right angles of Rt△, and C is the hypotenuse).

2、r=ab/ (a+b+c)

Sector inscribed circle

The circle tangent to the arc of the sector is called the inscribed circle of the sector.

The center O' of the inscribed circle is on the bisector of the central angle AOB of the sector.

OO'=R-r(R is the radius of the sector and R is the radius of the inscribed circle)

O' such as O'A⊥OA, vertical foot A, right triangle OAO'

∠O′OA = 30，O′A = R，OO′= R-R

∴r=(R-r)*sin30，r= 1/2，R=3r

Area of inscribed circle = π r 2,

The sector area is 60/360=65438+ 0/6 of the original circle area.

∴ Sector area = π r 2/6 = π (3r) 2/6 = 3π r 2/2

The ratio of the inscribed circle area to the sector area of the V shape is π r 2: (3 π r 2/2) = 2: 3.

Radius of inscribed circle of right triangle = 1/2 × (right side+other right side-hypotenuse)

The radius of the inscribed circle is r=2S÷C, where s represents the area of the triangle and c represents the perimeter of the triangle.

The inscribed circle is equal to 1 of the circumscribed circle.

The ratio of area to square is π: 4.