Schiffler point

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:$left\; [frac\{1\}\{cos\; B\; +\; cos\; C\},\; frac\{1\}\{cos\; C\; +\; cos\; A\},\; frac\{1\}\{cos\; A\; +\; cos\; B\}\; ight]$or, equivalently,:$left\; [frac\{b+c-a\}\{b+c\},\; frac\{c+a-b\}\{c+a\},\; frac\{a+b-c\}\{a+b\}\; ight]$where "a", "b", and "c" denote the side lengths of triangle "ABC".

References

*cite journal
author = Emelyanov, Lev; Emelyanova, Tatiana
title = A note on the Schiffler point
journal = Forum Geometricorum
volume = 3
year = 2003
pages = 113–116
id = MathSciNet | id = 2004116
url = http://forumgeom.fau.edu/FG2003volume3/FG200312index.html

*cite journal
author = Hatzipolakis, Antreas P.; van Lamoen, Floor; Wolk, Barry; Yiu, Paul
title = Concurrency of four Euler lines
journal = Forum Geometricorum
volume = 1
year = 2001
pages = 59–68
id = MathSciNet | id = 1891516
url = http://forumgeom.fau.edu/FG2001volume1/FG200109index.html

*cite journal
author = Nguyen, Khoa Lu
title = On the complement of the Schiffler point
journal = Forum Geometricorum
volume = 5
year = 2005
pages = 149–164
id = MathSciNet | id = 2195745
url = http://forumgeom.fau.edu/FG2005volume5/FG200521index.html

*cite journal
author = Schiffler, Kurt; Veldkamp, G. R.; van der Spek, W. A.
title = Problem 1018
year = 1985
journal = Crux Mathematicorum
volume = 11
pages = 51 Solution, vol. 12, pp. 150–152.

*cite journal
author = Thas, Charles
title = On the Schiffler center
journal = Forum Geometricorum
volume = 4
year = 2004
pages = 85–95
id = MathSciNet | id = 2081772
url = http://forumgeom.fau.edu/FG2004volume4/FG200412index.html

External links

* [http://www.gogeometry.com/geometry/schiffler_point_euler_line.htm Schiffler Point] with interactive dynamic geometry.