Difference between revisions of "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. A quadrilaterals that does have an incircle is called a Tangential Quadrilateral.

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 $\frac{ab}{a+b+c}=\frac{a+b-c}{2}$ .
For any polygon with an incircle, $K=sr$ , where $K$ is the area, $s$ is the semiperimeter, and $r$ is the inradius.

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 [[Image:Incenter.PNG|left|thumb|300px|Triangle ''ABC'' with [[incenter]] ''I'', with [[angle bisector]]s (red), [[incircle]] (blue), and [[inradius|inradii]] (green)]]
-An '''incircle''' of a [[convex]] [[polygon]] is a [[circle]] which is inside the figure and [[tangent line | 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 (geometry) | square]] [[rectangle]]s) do not have an incircle. Quadrilaterals that do have an incircle are called [[Tangential Quadrilaterals]].
+An '''incircle''' of a [[convex]] [[polygon]] is a [[circle]] which is inside the figure and [[tangent line | 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 (geometry) | square]] [[rectangle]]s) do not have an incircle. A quadrilaterals that does have an incircle is called a [[Tangential Quadrilateral]].
 ==Formulas==