Question 192
For a trapezoid $ABCD$ with $AB // CD$, suppose that $A$, $B$, $C$, $D$ lie in the clockwise direction. Let $\Gamma_1$ be the circle centered at $A$, and passing through $B$. Let $\Gamma_2$ be the circle centered at $C$, and passing through $D$. Let $P$ be the intersection (distinct from $B$, $D$) of the line $BD$ and the circle $\Gamma_1$. Let $\Gamma$ be the circle with the diameter $PD$. Let $X$ be the intersection (distinct from $P$) of $\Gamma$ and $\Gamma_1$. Let $Y$ be the intersection (distinct from $D$) of $\Gamma$ and $\Gamma_2$. Let $Q$ be the intersection of $\Gamma_2$ and the circumcircle of $XBY$. Show that $B$, $D$, $Q$ are collinear.