This is another problem from the Futility Closet website. It turned out to be pretty simple. The idea is to show the length of BC remains the same no matter where A is chosen on its arc of C1.
(Update 7/1/2020) There is more to this problem than I realized, thanks to a revisit prompted by a question from Deb Jyoti Mitra. See the revised Keyhole Problem.
Another good source of problems is the Futility Closet site. This puzzle involved finding the line of maximal length passing through the intersection of two circles. I solved it before looking at the Futility Closet solution. Their solution of course was short, sweet, and elegant. Mine was more like the old adage of cracking a walnut with a sledge hammer. Still, I thought there were some unexplained parts to the elegant solution that justified the effort on mine. At least my solution provided an interesting, though convoluted, alternative. See the Two Circles Puzzle.