Triangle △ABC
Lines joining the midpoints of each angle bisector to the vertices of △ABC
Lines perpendicular to each angle bisector at their midpoints
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.[1]
Coordinates
Trilinear coordinates for the Schiffler point are
-
1
cos
B
+
cos
C
:
1
cos
C
+
cos
A
:
1
cos
A
+
cos
B
{\displaystyle {\frac {1}{\cos B+\cos C}}:{\frac {1}{\cos C+\cos A}}:{\frac {1}{\cos A+\cos B}}}
[1]
or, equivalently,
-
b
+
c
−
a
b
+
c
:
c
+
a
−
b
c
+
a
:
a
+
b
−
c
a
+
b
{\displaystyle {\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. Archived from the original on July 6, 2003.
Further reading
- 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. Archived from the original on 2007-01-15. Retrieved 2007-01-17.
- 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. Archived from the original on 2007-03-19. Retrieved 2007-01-17.