Question 82
Let $\Omega$ be a circle, $S$ be a point outside $\Omega$ and $SA, SB$ be tangents from $S$ to $\Omega$. Arbitrary point $K$ is chosen on the segment $AB$. Points $U, W, R$ and $T$ are chosen on $\Omega$ such that $UW \cap RT = K$ and the points $W, T$ lie on the same half-plane with respect to the line $AB$. The line $SK$ intersects $\Omega$ at $P$ and $Q$ and $P$ lies on the same half-plane to the line $AB$ as $W$ and $T$. The lines $WR$ and $TU$ intersect $AB$ at $C$ and $D$ respectively. Let $M$ be the midpoint of $PQ$.
Prove that $\angle AMC = \angle BMD$.
Prove that $\angle AMC = \angle BMD$.