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

Solution: If $\ol{AB} \parallel \ol{CD}$ there is nothing to prove.
Otherwise let $X = \ol{AB} \cap \ol{CD}$.
Then the $Q_1$ and $Q_2$ are the isogonal conjugates of $P$
with respect to triangles $XBC$ and $XAD$.
Thus $X$, $Q_1$, $Q_2$ are collinear,
on the isogonal of $\ol{XP}$ with respect to
$\angle DXA = \angle CXB$.