Problem

Source: Romania JBMO TST 2024 Day 1 P4

Tags: geometry



Let $ABC$ be a triangle. An arbitrary circle which passes through the points $B,C$ intersects the sides $AC,AB$ for the second time in $D,E$ respectively. The line $BD$ intersects the circumcircle of the triangle $AEC$ at $P{}$ and $Q{}$ and the line $CE$ intersects the circumcircle of the triangle $ABD$ at $R{}$ and $S{}$ such that $P{}$ is situated on the segment $BD{}$ and $R{}$ lies on the segment $CE.$ Prove that: The points $P,Q,R$ and $S{}$ are concyclic. The triangle $APQ$ is isosceles. Petru Braica