Question 183
Given an oblique triangle $ABC$ with $BC \gt AC \gt AB$. Let $P_1 \neq P_2$ be two points on the plane such that, for $i = 1, 2$, if $AP_i, BP_i$, and $CP_i$ intersect the circumcircle of $\triangle ABC$ 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 $P_1P_2$ intersect the circumcircle of $\triangle ABC$ at $Q_1$ and $Q_2$. The two Simson lines of $Q_1$ and of $Q_2$ with respect to $\triangle ABC$ intersect at $W$.
Prove that $W$ lies on the nine-point circle of $\triangle ABC$.
Prove that $W$ lies on the nine-point circle of $\triangle ABC$.