Schiffler point

Point defined as a triangle center From Wikipedia, the free encyclopedia

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).

Diagram of the Schiffler point on an arbitrary triangle — Diagram of the Schiffler point

  Triangle $△ ABC$

  Angle bisectors; concur at incenter $I$

  Lines joining the midpoints of each angle bisector to the vertices of $△ ABC$

  Lines perpendicular to each angle bisector at their midpoints

  Euler lines; concur at the Schiffler point $Sp$

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

{\frac {1}{\cos B+\cos C}}:{\frac {1}{\cos C+\cos A}}:{\frac {1}{\cos A+\cos B}}

^[1]

or, equivalently,

{\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

Further reading

External links

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