Sharygin 2008, final round, grade 9, problem 8
Sharygin 2008, final round, grade 9, problem 8 is a plane geometry problem from the Sharygin Geometry Olympiad 2008, final round, grade 9, problem 8: the perpendicular bisector of PQ and three concurrent lines.
Problem
Points $P$, $Q$ lie on the circumcircle $\omega$ of triangle $ABC$. The medial perpendicular $\ell$ to $PQ$ intersects $BC$, $CA$, $AB$ in points $A'$, $B'$, $C'$. Let $A''$, $B''$, $C''$ be the second common points of $\ell$ with the circles $A'PQ$, $B'PQ$, $C'PQ$. Prove that $AA''$, $BB''$, $CC''$ concur.
Source Sharygin Geometry Olympiad 2008, final round, grade 9, problem 8
The diagram
Drag any point and the whole construction follows; ▶ runs a point along its circle and back. Tap a point to pin it, double-click to go back to the start. Effects redraws the figure in grayscale for print, in colour, or with the givens in focus.