Question 2
Let $ABC$ be an acute-angled triangle with $AB \lt AC$. Denote its circumcircle by $\Omega$ and denote the midpoint of $\operatorname{arc} CAB$ by $S$. Let the perpendicular from $A$ to $BC$ meet $BS$ and $\Omega$ at $D$ and $E \neq A$ respectively. Let the line through $D$ parallel to $BC$ meet line $BE$ at $L$ and denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$.
Prove that the line tangent to $\omega$ at $P$, and line $BS$ intersect on the internal bisector of $\angle BAC$.
Prove that the line tangent to $\omega$ at $P$, and line $BS$ intersect on the internal bisector of $\angle BAC$.