Schiffler point

Diagram of the Schiffler Point

  Triangle △ABC

  Angle bisectors; concur at incenter I

  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).

Definition

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.

Trilinear coordinates for the Schiffler point are

${\frac {1}{\cos B+\cos C}}:{\frac {1}{\cos C+\cos A}}:{\frac {1}{\cos A+\cos B}}$

or, equivalently,

${\frac {b+c-a}{b+c}}:{\frac {c+a-b}{c+a}}:{\frac {a+b-c}{a+b}}$

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

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 (1985). "Problem 1018" (PDF). Crux Mathematicorum. 11: 51. Retrieved September 24, 2023.
• Veldkamp, G. R. & van der Spek, W. A. (1986). "Solution to Problem 1018" (PDF). Crux Mathematicorum. 12: 150–152. Retrieved September 24, 2023.
• Thas, Charles (2004). "On the Schiffler center". Forum Geometricorum. 4: 85–95. MR 2081772.

External links

• Weisstein, Eric W. "Schiffler Point". MathWorld.