Schiffler point

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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}\right]

or, equivalently,

\left[\frac{b+c-a}{b+c}, \frac{c+a-b}{c+a}, \frac{a+b-c}{a+b}\right]

where a, b, and c denote the side lengths of triangle ABC.

[edit] References

  • Schiffler, Kurt; Veldkamp, G. R.; van der Spek, W. A. (1985). "Problem 1018". Crux Mathematicorum 11: 51.  Solution, vol. 12, pp. 150–152.

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