Problem

Source: Iranian TST 2018, third exam day 1, problem 1

Tags: geometry, Iran, Iranian TST



Two circles $\omega_1(O)$ and $\omega_2$ intersect each other at $A,B$ ,and $O$ lies on $\omega_2$. Let $S$ be a point on $AB$ such that $OS\perp AB$. Line $OS$ intersects $\omega_2$  at $P$ (other than $O$). The bisector of $\hat{ASP}$ intersects  $\omega_1$ at $L$ ($A$ and $L$ are on the same side of the line $OP$). Let $K$ be a point on $\omega_2$ such that $PS=PK$ ($A$ and $K$ are on the same side of the line $OP$). Prove that $SL=KL$. Proposed by Ali Zamani