Incircle

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Triangle ABC with incenter I, with angle bisectors (red), incircle (blue), and inradii (green)

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.

Formulas

The radius of an incircle of a triangle (the inradius) with sides $a,b,c$ and area $K$ is $\frac{2K}{a+b+c}$
The radius of an incircle of a right triangle (the inradius) with legs $a,b$ and hypotenuse $c$ is $\frac{ab}{a+b+c}$

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Incircle

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Formulas