Question 69
Let $ABC$ be a triangle. An arbitrary circle which passes through the points $B$ and $C$ intersects the sides $AC$ and $AB$ for the second time in $D$ and $E$, respectively. The line $BD$ intersects the circumcircle of 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:
a) the points $P, Q, R$ and $S$ are concyclic;
b) the triangle $APQ$ is isosceles.
a) the points $P, Q, R$ and $S$ are concyclic;
b) the triangle $APQ$ is isosceles.