CN-TST 2022 Problem 8

Given a non-right triangle $ABC$ with $BC>AC>AB$ . Two points $P_1 \neq P_2$ on the plane satisfy that, for $i=1,2$ , if $AP_i, BP_i$ and $CP_i$ intersect the circumcircle of the triangle $ABC$ at $D_i, E_i$ , and $F_i$ , respectively, then $D_iE_i \perp D_iF_i$ and $D_iE_i = D_iF_i \neq 0$ . Let the line $P_1P_2$ intersects the circumcircle of $ABC$ at $Q_1$ and $Q_2$ . The Simson lines of $Q_1$ , $Q_2$ with respect to $ABC$ intersect at $W$ .

Prove that $W$ lies on the nine-point circle of $ABC$ .