Incenter–excenter lemma

The incenter–excenter lemma is a triangle-geometry result relating a triangle’s incenter to one or more of its excenters. Its exact statement varies across sources.

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Incenter: The point where a triangle’s three internal angle bisectors meet, equidistant from its sides. It is the interior center named in the lemma.
Barycentric coordinates: A coordinate system for points in a triangle, expressed through weighted combinations of its vertices. They give compact coordinates for the incenter and excenters.
Triangle center: A point defined from a triangle by a geometric construction or invariant property. The incenter and excenters are members of this broader family.
Circumcenter: The point equidistant from a triangle’s three vertices, where its perpendicular bisectors meet. Unlike the incenter and excenters, it is defined by vertex distances.
Excenter: One of three triangle centers formed by intersecting one internal and two external angle bisectors. The lemma relates the incenter to these exterior-associated centers.
Trilinear coordinates: Triangle coordinates in which a point is represented by its directed distances from the three sidelines. Equal-distance descriptions of the centers become direct coordinate conditions.
Ceva’s theorem: A criterion determining when three lines through a triangle’s vertices meet at one point. It can establish concurrence of angle-bisector lines in center proofs.
Orthocenter: The point where a triangle’s three altitudes meet. It arises from perpendicularity rather than angle-bisector distances.
Angle bisector: A ray or line that divides an angle into two equal angles. The incenter and excenters are defined by intersections of angle bisectors.
Angle-bisector theorem: A theorem stating that an angle bisector divides the opposite side in the ratio of the adjacent sides. It converts bisector geometry into side-length relations.