Perpendiculars to the harmonic lines are also harmonic lines
Source: Kazakhstan National Olympiad 2024 (10-11 grade), P6
March 21, 2024
geometry
Problem Statement
The circle with center at point inscribed in an triangle () touches the sides , and at points , and , respectively. The circumcircles of triangles and intersect secondary at point The lines and intersect at point and intersects the line at points and , respectively. The tangent lines to , other than , passing through points and touch at points and , respectively. Let the lines and intersect at the point . Prove that if is a midpoint of segment then .