Question 170
Let $ABCD$ be a cyclic quadrangle and let $M$ and $N$ be the midpoints of the sides $AD$ and $BC$, respectively. The circle through $A$ and $D$ tangent to $AC$ crosses the line $AB$ again at $P$ and the circle through $B$ and $C$ tangent to $BD$ crosses the line $AB$ again at $Q$. Let $t_A$ be the tangent of the circle $AMP$ at $A$ and let $t_B$ be the tangent of the circle $BNQ$ at $B$. Prove that the lines $CD$, $t_A$ and $t_B$ are concurrent.