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Schiffler point - Wikipedia

  • ️Mon Jan 15 2007

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Diagram of the Schiffler point on an arbitrary triangle
Diagram of the Schiffler Point

  Triangle △ABC

  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

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.[1]

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.

  1. ^ a b Emelyanov, Lev; Emelyanova, Tatiana (2003). "A note on the Schiffler point". Forum Geometricorum. 3: 113–116. MR 2004116. Archived from the original on July 6, 2003.