Reuschle's theorem

Reuschle's theorem is a classical result in triangle geometry, attributed to Karl Gustav Reuschle, concerning relationships among a triangle’s centers and associated circles.

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Triangle center: A point defined by a geometric construction or property of a triangle. The theorem relates distinguished points associated with a triangle.
Euler's theorem on the circumradius and inradius: The relation stating that the squared distance between a triangle’s circumcenter and incenter equals R(R−2r). It is another classical relation tying triangle centers to radii.
Karl Gustav Reuschle: A German mathematician and geographer associated with a classical theorem in triangle geometry. The theorem is attributed to him.
Dynamic geometry software: Software that lets users construct geometric figures and manipulate them while preserving defined relationships. It can illustrate how a proposed center-circle relation behaves as a triangle changes.
Circumcircle: The unique circle passing through all three vertices of a nondegenerate triangle. Triangle-circle relationships provide the setting for the theorem.
Euler line: The line containing a triangle’s circumcenter, centroid, and orthocenter. It is a central alignment result in the same geometry of triangle centers.
Triangle geometry: The study of points, lines, circles, and other configurations defined by triangles. Reuschle’s theorem belongs to this field.
Geometric construction: The creation of geometric objects from specified data using defined construction operations. Triangle centers and their associated circles are defined through constructions.
Incircle: The circle inside a triangle tangent to all three of its sides. It is one of the classical circles used in triangle-center relations.
Nine-point circle: The circle through a triangle’s side midpoints, altitude feet, and the midpoints between vertices and orthocenter. This associated circle is a common participant in center-and-circle configurations.