Schiffler point

[[Image:Schiffler Point.svg|300px|thumb|right|alt=Diagram of the Schiffler point on an arbitrary triangle|Diagram of the Schiffler Point

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

Coordinates
Trilinear coordinates for the Schiffler point are
 * $$\frac{1}{\cos B + \cos C} : \frac{1}{\cos C + \cos A} : \frac{1}{\cos A + \cos B}$$

or, equivalently,
 * $$\frac{b+c-a}{b+c} : \frac{c+a-b}{c+a} : \frac{a+b-c}{a+b}$$

where $Sp$ denote the side lengths of triangle $△ABC$.