Question 5
Let $P$ be a point inside convex quadrilateral $ABCD$.
Points $Q_1$ and $Q_2$ are located within $ABCD$ such that
\[\begin{aligned}
\angle Q_1BC=\angle ABP, & \qquad \angle Q_1CB=\angle DCP, \\
\angle Q_2AD=\angle BAP, & \qquad \angle Q_2DA=\angle CDP.
\end{aligned}\]
Prove that $\overline{Q_1Q_2} \parallel \overline{AB}$
if and only if $\overline{Q_1Q_2} \parallel \overline{CD}$.
Points $Q_1$ and $Q_2$ are located within $ABCD$ such that
\[\begin{aligned}
\angle Q_1BC=\angle ABP, & \qquad \angle Q_1CB=\angle DCP, \\
\angle Q_2AD=\angle BAP, & \qquad \angle Q_2DA=\angle CDP.
\end{aligned}\]
Prove that $\overline{Q_1Q_2} \parallel \overline{AB}$
if and only if $\overline{Q_1Q_2} \parallel \overline{CD}$.