Feuerbach's theorem

Feuerbach's theorem states that a triangle's nine-point circle is internally tangent to its incircle and externally tangent to each of its three excircles.

Nine-point circle: A triangle's circle through its three side midpoints, three altitude feet, and three Euler points. This is the circle whose tangencies the theorem asserts.
Nine-point center: The center of a triangle's nine-point circle, midway between its circumcenter and orthocenter. Its distance to the incenter satisfies the relation behind the incircle tangency.
Karl Wilhelm Feuerbach: A German mathematician who studied triangle geometry and proved the theorem now bearing his name. He established the incircle and excircle tangency result.
Feuerbach point: The point where a triangle's nine-point circle touches its incircle. It names the unique point of internal tangency in the theorem.
Equilateral triangle: A triangle with three equal sides and three equal angles. Its centers coincide in highly symmetric ways, making the tangencies easy to inspect.
Incircle: The circle inside a polygon that is tangent to every side. Feuerbach's theorem identifies its tangency with the nine-point circle.
Euler's triangle theorem: The relation between a triangle's circumradius, inradius, and distance between circumcenter and incenter: d²=R(R−2r). Its radius relation helps establish the incenter-to-nine-point-center distance.
Journal für die reine und angewandte Mathematik: A mathematics journal founded by August Leopold Crelle in 1826. Feuerbach published his triangle-geometry work in this journal.
Feuerbach hyperbola: A rectangular hyperbola associated with a triangle and its Feuerbach point. It extends the theorem's configuration into a locus-based triangle result.
Isosceles triangle: A triangle with at least two equal sides. Its symmetry reduces the four tangency configurations to fewer distinct cases.

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Reuschle's theoremRelated: It exemplifies a classical theorem relating a triangle’s centers and associated circles.