Schiffler point

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

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.

Coordinates

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". MR 2004116
.

Hatzipolakis, Antreas P.; van Lamoen, Floor; Wolk, Barry; Yiu, Paul (2001). "Concurrency of four Euler lines". MR 1891516
.

Nguyen, Khoa Lu (2005). "On the complement of the Schiffler point". 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". MR 2081772
.

External links

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