RIOPLATENSE 2014 Problem 5

In the segment $A C$ a point $B$ is taken. Construct circles $T_1, T_2$ and $T_3$ of diameters $A B, BC$ and $AC$ respectively. A line that passes through $B$ cuts $T_3$ in the points $P$ and $Q$ , and the circles $T_1$ and $T_2$ respectively at points $R$ and $S$ . Prove that $PR = Q S$ .