Schiffler point explained

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

Coordinates

Trilinear coordinates for the Schiffler point are

(1)/(\cos B + \cos C) : (1)/(\cos C + \cos A) : (1)/(\cos A + \cos B) or, equivalently,(b + c - a)/(b + c) : (c + a - b)/(c + a) : (a + b - c)/(a + b) where denote the side lengths of triangle .

References

• Emelyanov, Lev . Emelyanova, Tatiana . A note on the Schiffler point . . 3 . 2003 . 113–116 . 2004116 .
• Hatzipolakis, Antreas P. . van Lamoen, Floor . Wolk, Barry . Yiu, Paul . Concurrency of four Euler lines . . 1 . 2001 . 59–68 . 1891516 .
• Nguyen, Khoa Lu . On the complement of the Schiffler point . . 5 . 2005 . 149–164 . 2195745 .
• Schiffler, Kurt . Problem 1018 . 1985 . . 11 . 51 . September 24, 2023 .
• Veldkamp, G. R. . van der Spek, W. A. . amp . Solution to Problem 1018 . 1986 . . 12 . 150–152 . September 24, 2023 .
• Thas, Charles . On the Schiffler center . . 4 . 2004 . 85–95 . 2081772 .