Schiffler point

Schiffler's point S as the intersection of the Euler lines e ₀ , e ₁ , e ₂ and e ₃

The Schiffler point is one of the special points of a triangle and has the Kimberling number X (21). If I is the center of the inscribed circle , the Euler lines of the triangles ABC, BCI, CAI and ABI intersect at one point. This intersection was introduced in 1985 by the toy manufacturer and amateur geometer Kurt Schiffler in the Canadian mathematics journal Crux Mathematicorum and is now known as the Schiffler point and the statement that all four Euler lines intersect at this point is Schiffler's theorem .

Coordinates

Schiffler point ( ) ${\displaystyle X_{21}}$
Trilinear coordinates	${\displaystyle {\frac {b+c-a}{b+c}}\,:\,{\frac {c+a-b}{c+a}}\,:\,{\frac {a+b-c}{a+b}}}$ ${\displaystyle ={\frac {1}{\cos \beta +\cos \gamma }}\,:\,{\frac {1}{\cos \gamma +\cos \alpha }}\,:\,{\frac {1}{\cos \alpha +\cos \beta }}}$
Barycentric coordinates	${\displaystyle {\frac {a(s-a)}{b+c}}\,:\,{\frac {b(s-b)}{c+a}}\,:\,{\frac {c(s-c)}{a+b}}}$ ${\displaystyle ={\frac {a}{\cos \beta +\cos \gamma }}\,:\,{\frac {b}{\cos \gamma +\cos \alpha }}\,:\,{\frac {c}{\cos \alpha +\cos \beta }}}$

Web links

• Eric W. Weisstein : Schiffler Point . In: MathWorld (English).

Individual evidence

1. ↑ Joe Goggins: The Converse of Schiffler's theorem . In: Crux Mathematicorum , with Mathematical Mayhem, Canadian Mathematical Society, 2007, Volume 33, No. 6, p. 354, cms.math.ca (PDF)
2. ↑ Kurt Schiffler: Problem 1018 . In: Crux Mathematicorum , Volume 11, No. 2, February 1985, p. 51, cms.math.ca (PDF). GR Veldkamp, ​​WA van der Spek: Solutions to Problem 1018 . In: Crux Mathematicorum , Volume 12, No. 6, June 1986, p. 151, cms.math.ca (PDF)