Question 18
Let $A B C$ be an acute-angled triangle with $A B\lt A C$. Denote its circumcircle by $\Omega$ and denote the midpoint of $\operatorname{arc} C A B$ by $S$. Let the perpendicular from $A$ to $B C$ meet $B S$ and $\Omega$ at $D$ and $E \neq A$ respectively. Let the line through $D$ parallel to $B C$ meet line $B E$ at $L$ and denote the circumcircle of triangle $B D L$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$.
Prove that the line tangent to $\omega$ at $P$, and line $B S$ intersect on the internal bisector of $\angle B A C$.
Prove that the line tangent to $\omega$ at $P$, and line $B S$ intersect on the internal bisector of $\angle B A C$.