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.

Trilinear coordinates for the Schiffler point are

$\left[\frac{1}{\cos B %2B \cos C}, \frac{1}{\cos C %2B \cos A}, \frac{1}{\cos A %2B \cos B}\right]$

or, equivalently,

$\left[\frac{b%2Bc-a}{b%2Bc}, \frac{c%2Ba-b}{c%2Ba}, \frac{a%2Bb-c}{a%2Bb}\right]$

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. http://forumgeom.fau.edu/FG2003volume3/FG200312index.html.

Hatzipolakis, Antreas P.; van Lamoen, Floor; Wolk, Barry; Yiu, Paul (2001). "Concurrency of four Euler lines". Forum Geometricorum 1: 59–68. MR 1891516. http://forumgeom.fau.edu/FG2001volume1/FG200109index.html.

Nguyen, Khoa Lu (2005). "On the complement of the Schiffler point". Forum Geometricorum 5: 149–164. MR 2195745. http://forumgeom.fau.edu/FG2005volume5/FG200521index.html.

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. http://forumgeom.fau.edu/FG2004volume4/FG200412index.html.