I'm going through Solution 1 for problem G2 from IMO 2006 (https://www.imo-official.org/problems/IMO2006SL.pdf).
In the penultimate paragraph they conclude that homothety $h$ takes circle $(ABP)$ to circle $(CDQ)$. Why is that? Why is point $S$ the homothetic center of these two circles? How does tangency of just one line $SB$ to these two circles lead to this?