incircle with center I of triangle ABC touches the side BC
Source: Vietnam TST 2003 for the 44th IMO, problem 2
June 26, 2005
geometrycircumcircleratiogeometric transformationreflectionanalytic geometryhomothety
Problem Statement
Given a triangle . Let be the circumcenter of this triangle . Let , , be the feet of the altitudes of triangle from the vertices , , , respectively. Denote by , , the midpoints of these altitudes , , , respectively. The incircle of triangle has center and touches the sides , , at the points , , , respectively. Prove that the four lines , , and are concurrent. (When the point concides with , we consider the line as an arbitrary line passing through .)