Incircle

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An incircle of a convex polygon is a circle which is inside the figure and tangent to each side. Every triangle and regular polygon has a unique incircle, but in general polygons with 4 or more sides (such as non- square rectangles) do not have an incircle. A quadrilateral that does have an incircle is called a Tangential Quadrilateral. For a triangle, the center of the incircle is the Incenter.

Formulas

The radius of an incircle of a triangle (the inradius) with sides $a,b,c$ and area $A$ is $\frac{2A}{a+b+c}$
The radius of an incircle of a right triangle (the inradius) with legs $a,b$ and hypotenuse $c$ is $r=\frac{ab}{a+b+c}=\frac{a+b-c}{2}$ .
For any polygon with an incircle, $A=sr$ , where $A$ is the area, $s$ is the semiperimeter, and $r$ is the inradius.
The coordinates of the incenter (center of incircle) are $(\dfrac{aA_x+bB_x+cC_x}{a+b+c}, \dfrac{aA_y+bB_y+cC_y}{a+b+c})$ , if the coordinates of each vertex are $A(A_x, A_y)$ , $B(B_x, B_y)$ , and $C(C_x, C_y)$ , the side opposite of $A$ has length $a$ , the side opposite of $B$ has length $b$ , and the side opposite of $C$ has length $c$ .

The formula for the semiperimeter is $s=\frac{a+b+c}{2}$ .

And area of the triangle by Heron is $A^2=s(s-a)(s-b)(s-c)$ .

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Incircle

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