Proof of concurrency of transverse tangents and line joining centers of two disjoint circles

Question

I read that "the transverse common tangents to two disjoint circles and the line joining the center of the circles are concurrent" and tried proving it.

Can I get a hint to prove that lines FI, GH and AC are concurrent at E?

Answer 1

In the figure, we can prove that $A, C, B$ are collinear by showing that

$(1)$ $\alpha = \frac{\angle JCI}{2}= \frac{\angle HCK}{2}=\beta$,

and hence

$(2)$ $\angle ACH+ \angle HCB=180^o$