Question 48
Let $P$ be a given point inside quadrilateral $ABCD$. Points $Q_1$ and $Q_2$ are located within $ABCD$ such that
$$
\angle Q_1 BC = \angle ABP, \quad \angle Q_1 CB = \angle DCP, \quad \angle Q_2 AD = \angle BAP, \quad \angle Q_2 DA = \angle CDP.
$$
Prove that $\overline{Q_1Q_2} \parallel \overline{AB}$ if and only if $\overline{Q_1Q_2} \parallel \overline{CD}$.
$$
\angle Q_1 BC = \angle ABP, \quad \angle Q_1 CB = \angle DCP, \quad \angle Q_2 AD = \angle BAP, \quad \angle Q_2 DA = \angle CDP.
$$
Prove that $\overline{Q_1Q_2} \parallel \overline{AB}$ if and only if $\overline{Q_1Q_2} \parallel \overline{CD}$.