SHARYGIN 2022 Problem 17

Let a point $P$ lie inside a triangle $ABC$ . The rays starting at $P$ and crossing the sides $BC$ , $AC$ , $AB$ under the right angle meet the circumcircle of $ABC$ at $A_{1}$ , $B_{1}$ , $C_{1}$ respectively. It is known that lines $AA_{1}$ , $BB_{1}$ , $CC_{1}$ concur at point $Q$ . Prove that all such lines $PQ$ concur.