$4.$ Two touching circles $S$ and $T$ share a common tangent which meets $S$ at $A$ and $T$ at $B$. Let $AP$ be a diameter of $S$ and let the tangent from $P$ to $T$ touch it at $Q$. Show that $AP = PQ$.
I'm unsure of the configuration. If $P$ is on the opposite side of the circle to $A$, does that not suggest the tangent to that point produces a line parallel to the other tangent. Then the other circle would have to be of equal size and the statement of $AP=PQ$ becomes trivial and I just don't see it being so easy. So to anyone who can come up with a hint or full solution if you like, thank you in advance.