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).

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

Definition

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]

Coordinates

Trilinear coordinates for the Schiffler point are

${\displaystyle {\frac {1}{\cos B+\cos C}}:{\frac {1}{\cos C+\cos A}}:{\frac {1}{\cos A+\cos B}}}$ ^[1]

or, equivalently,

${\displaystyle {\frac {b+c-a}{b+c}}:{\frac {c+a-b}{c+a}}:{\frac {a+b-c}{a+b}}}$

where a, b, c denote the side lengths of triangle △ABC.