Proof of the Arbelos Theorem

Recently I posted an account of the beautiful Arbelos Theorem found over two thousand years back by the great Archimedes of Syracuse. (See Archimedes's Arbelos Theorem.) Here I give a link to the original proof given by Archimedes:

• Proof by Archimedes of Arbelos Theorem

It is a "pure geometry" proof - it involves almost no computation, and it skillfully uses results from circle geometry and the geometry of homothetic transformations. (Comment. The word "homothety" is of relatively recent origin, but the notion of a homothety or enlargement about a point is an old one.)

However it establishes only that circles $\omega_4$ and $\omega_5$ have equal radius. I do not know any such proof that circle $\omega_7$ too has the same radius.My proof is computational, and I will post it shortly.