Point defined as a triangle center
Diagram of the Schiffler Point 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.
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}}}
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 . [permanent dead link ] 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 .
External links
