Question 53
Let $p$ and $q$ be two parallel lines. Circle $k$ touches the line $p$ at $A$ and intersects $q$ at two different points, $B$ and $C$. Let $T$ be some point on $p$. Segments $\overline{TB}$ and $\overline{TC}$ intersect the shorter arc $\widehat{AC}$ at $K$ and $L$ respectively. Points $K$ and $L$ are both different from $B$ and $C$.
Prove that the line $KL$ passes through the midpoint of the segment $\overline{AT}$.
Prove that the line $KL$ passes through the midpoint of the segment $\overline{AT}$.