Question 34
Let $AB$ be a diameter of a circle $\Omega$ and $P$ be any point not on the line through $A$ and $B$. Suppose the line through $P$ and $A$ cuts $\Omega$ again in $U$, and the line through $P$ and $B$ cuts $\Omega$ again in $V$. (Note that in case of tangency $U$ may coincide with $A$ or $V$ may coincide with $B$. Also, if $P$ is on $\Omega$ then $P=U=V$.) Suppose that $|PU|=s|PA|$ and $|PV|=t|PB|$ for some nonnegative real numbers $s$ and $t$. Determine the cosine of the angle $APB$ in terms of $s$ and $t$.