Schiffler point

In geometry, the Schiffler point of a triangle is a point defined from the triangle that is invariant 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, and ABC. Schiffler's theorem states that these lines are concurrent.

Trilinear coordinates for the Schiffler point are

$\left[\frac{1}{\cos B + \cos C}, \frac{1}{\cos C + \cos A}, \frac{1}{\cos A + \cos B}\right]$

or, equivalently,

$\left[\frac{b+c-a}{b+c}, \frac{c+a-b}{c+a}, \frac{a+b-c}{a+b}\right]$

where a, b, and c denote the side lengths of triangle ABC.

References

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