Question 19
In the plane, a triangle $ABC$ and a point $P$ are given. Let the circumcentres of $\triangle ABC$, $\triangle BPC$, $\triangle CPA$, $\triangle APB$ be the points $O$, $D$, $E$, $F$, respectively. Let the lines $BC$ and $EF$ meet at the point $T$, and let $X$ be the reflection of the point $O$ in the line $EF$. Prove that $PT \perp DX$.