Schiffler point
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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 + \cos C}, \frac{1}{\cos C + \cos A}, \frac{1}{\cos A + \cos B}\right]](../../../math/f/3/8/f380ff7ee7b6189331af28a7cf1e7132.png)
or, equivalently,
![\left[\frac{b+c-a}{b+c}, \frac{c+a-b}{c+a}, \frac{a+b-c}{a+b}\right]](../../../math/7/c/f/7cf190e723fb2d4751dfbd58a3257fac.png)
where a, b, and c denote the side lengths of triangle ABC.
[edit] References
- Emelyanov, Lev; Emelyanova, Tatiana (2003). "A note on the Schiffler point". Forum Geometricorum 3: 113–116. MR2004116.
- Hatzipolakis, Antreas P.; van Lamoen, Floor; Wolk, Barry; Yiu, Paul (2001). "Concurrency of four Euler lines". Forum Geometricorum 1: 59–68. MR1891516.
- Nguyen, Khoa Lu (2005). "On the complement of the Schiffler point". Forum Geometricorum 5: 149–164. MR2195745.
- 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. MR2081772.
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