# 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).

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

## Coordinates

Trilinear coordinates for the Schiffler point are

${\displaystyle \left[{\frac {1}{\cos B+\cos C}},{\frac {1}{\cos C+\cos A}},{\frac {1}{\cos A+\cos B}}\right]}$

or, equivalently,

${\displaystyle \left[{\frac {b+c-a}{b+c}},{\frac {c+a-b}{c+a}},{\frac {a+b-c}{a+b}}\right]}$

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