Consider the tangents from O1 to (O2) and O2 to (O1) intersecting at H.
Create J in such a way that triangle JO1O2 has H as its incenter.
The incircle of this triangle meets the sides JO1 and JO2 in P and Q respectively.
The circle with diameter PQ is Archimedean circle (A62).
Note that if JO1 meets (O1) in K and the circle with diameter CK is Archimedean circle (A50a).
Similarly if JO2 meets (O1) in L, and the circle with diameter CL is Archimedean circle (A50b).
[Miguel Ochoa Sanchez (conjecture) and Emmanuel Antonio José García (proof), june 2014]
Source: A. Bogomolny, Shedding Light on the Ball for Eyeballing from Interactive Mathematics Miscellany and Puzzles http://www.cut-the-knot.org/pythagoras/LightingTheBall.shtml
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