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

The incenter of a triangle is the intersection of its (interior) angle bisectors. The incenter is the center of the incircle. Every nondegenerate triangle has a unique incenter.

Proof of Existence

Consider a triangle $\displaystyle ABC$. Let $\displaystyle I$ be the intersection of the respective interior angle bisectors of the angles $\displaystyle BAC$ and $\displaystyle CBA$. We observe that since $\displaystyle I$ lies on an angle bisector of $\displaystyle BAC$, is equidistant from $\displaystyle AB$ and $\displaystyle CA$; likewise, it is equidistant from $\displaystyle BC$ and $\displaystyle AB$; hence it is equidistant from $\displaystyle BC$ and $\displaystyle BC$ and $\displaystyle CA$ and therefore lies on an angle bisector of $\displaystyle ACB$. Since it lies within the triangle $\displaystyle ABC$, this is the interior angle bisector of $\displaystyle ACB$. Since $\displaystyle I$ is equidistant from all three sides of the triangle, it is the incenter.

It should be noted that this proof parallels that for the existance of the circumcenter.

The proofs of existance for the excenters is the same, except that certain angle bisectors are exterior.

Properties of the Incenter

$\bullet$ The incenter of any triangle lies within the orthocentroidal circle.

$\bullet$ The unnormalised areal coordinates of the incenter are $(a,b,c)$

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