Schiffler point

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 Template:Math with the incenter Template:Mvar has its Schiffler point at the point of concurrence of the Euler lines of the four triangles Template:Math. Schiffler's theorem states that these four lines all meet at a single point.

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 Template:Mvar denote the side lengths of triangle Template:Math.

Further reading

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External links

• Template:MathWorld