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.

Trilinear coordinates for the Schiffler point are

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

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.

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• Schiffler, Kurt (1985). "Problem 1018" (PDF). Crux Mathematicorum. 11: 51. Retrieved September 24, 2023.
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• Weisstein, Eric W. "Schiffler Point". MathWorld.