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Proof of Existence
Consider a triangle . Let be the intersection of the respective interior angle bisectors of the angles and . We observe that since lies on an angle bisector of , is equidistant from and ; likewise, it is equidistant from and ; hence it is equidistant from and and and therefore lies on an angle bisector of . Since it lies within the triangle , this is the interior angle bisector of . Since is equidistant from all three sides of the triangle, it is the incenter.
It should be noted that this proof parallels that for the existence of the circumcenter.
The proofs of existence for the excenters is the same, except that certain angle bisectors are exterior.
Properties of the Incenter
The incenter of any triangle lies within the orthocentroidal circle.
The unnormalised areal coordinates of the incenter are
Let be a point on the circumcircle of such that bisects . Then points , , and lie on a circle centered at .