USAMO 2018 Problem 5

In convex cyclic quadrilateral $ABCD$ , we know that lines $AC$ and $BD$ intersect at $E$ , lines $AB$ and $CD$ intersect at $F$ , and lines $BC$ and $DA$ intersect at $G$ . Suppose that the circumcircle of $\triangle ABE$ intersects line $CB$ at $B$ and $P$ , and the circumcircle of $\triangle ADE$ intersects line $CD$ at $D$ and $Q$ , where $C,B,P,G$ and $C,Q,D,F$ are collinear in that order. Prove that if lines $FP$ and $GQ$ intersect at $M$ , then $\angle MAC = 90^\circ$ .

Proposed by Kada Williams