Let a line $m$ touch the incircle of triangle $ABC$. The lines passing through the incenter $I$ and perpendicular to $AI, BI, CI$ meet $m$ at points $A', B', C'$ respectively. Prove that $AA', BB'$ and $CC'$ concur.
Problem
Source: Sharygin Geometry Olympiad 2017 First Round P23 grades 10-11
Tags: geometry, concurrency, concurrent, perpendicular lines