TSTST 2026 Problem 1

Let $ABC$ be an acute scalene triangle. Let $O$ be the circumcenter of $ABC$, $D$ be the foot of the altitude from $A$ to $BC$, and $P$ be a point on line $AO$ not lying on the circumcircle of $ABC$. The circle with diameter $AP$ intersects the circumcircle of $BCP$ at $E \neq P$, and the line through $A$ parallel to $BC$ at $F \neq A$. Prove that $D$, $E$, and $F$ are collinear.