Schiffler point

In geometry, the Schiffler point of a triangle is a triangle center, a point defined from the triangle that is equivariant under Euclidean transformations of the triangle.

This point was first defined and investigated by Schiffler et al. (1985).

A triangle △ABC with the incenter I has its Schiffler point at the point of concurrence of the Euler lines of the four triangles △BCI, △CAI, △ABI, △ABC.

Schiffler's theorem states that these four lines all meet at a single point.

[1] Trilinear coordinates for the Schiffler point are or, equivalently, where a, b, c denote the side lengths of triangle △ABC.

Diagram of the Schiffler point on an arbitrary triangle
Diagram of the Schiffler Point
Triangle △ ABC
Angle bisectors ; concur at incenter I
Lines joining the midpoints of each angle bisector to the vertices of △ ABC
Lines perpendicular to each angle bisector at their midpoints
Euler lines ; concur at the Schiffler point Sp