Hardcore triangle geometry [A', B', C' on OI]
Source: 239MO 2000, classes 10-11, problem 8, by M. Sonkin, extended
October 7, 2004
geometryincentercircumcircleEulerangle bisectorperpendicular bisectorgeometry proposed
Problem Statement
The perpendicular bisectors of the sides AB and BC of a triangle ABC meet the lines BC and AB at the points X and Z, respectively. The angle bisectors of the angles XAC and ZCA intersect at a point B'. Similarly, define two points C' and A'. Prove that the points A', B', C' lie on one line through the incenter I of triangle ABC.
Extension: Prove that the points A', B', C' lie on the line OI, where O is the circumcenter and I is the incenter of triangle ABC.
Darij