Question 143
Let $ABCD$ be a quadrilateral inscribed in a circle. Let $\Delta_{1}$ be the line parallel to $(BC)$ through $A$ and $\Delta_{2}$ the line parallel to $(AD)$ through $B$. Let $E$ be the intersection point of $\Delta_{1}$ with $(CD)$ and $F$ the intersection point of $\Delta_{2}$ with $(CD)$. The line perpendicular to $\Delta_{1}$ through $A$ meets $(BC)$ at $P$, and the line perpendicular to $\Delta_{2}$ through $B$ meets $(AD)$ at $Q$. Let $\Gamma_{1}$ be the circumcircle of triangle $ADE$ and $\Gamma_{2}$ the circumcircle of triangle $BFC$. Show that $\Gamma_{1}$ and $\Gamma_{2}$ are tangent if and only if $(DP)$ and $(CQ)$ are perpendicular.