The proof is trivial, using coordinates with origin at O. The following geometric proof is due to Antreas Varverakis. Consider the circles d = (C',C'H), e = (C'',C''A), which are symmetric of (b) with respect to the lines OG and OF respectively. Obviously G and F are radical centers of the tripples of circles: ((b),(d),(c)) and ((b),(e),(c)) respectively. Since, in each tripple, the two last circles are fixed, the lines [LG] and [KF] are also fixed.

The line [FG] envelopes an ellipse with one focus at O. Look at ThreeCirclesProblem1.html if you are interested in.

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