I guess the Ptolemy hint did the trick. Or maybe with just the figure, you'd have figured it out too...
There are some beautiful other problems in Math Horizons, I'll have to take some time to curate the ones that are easy to post here.
What makes this so tricky is that there is no right triangle involved, yet you look for one because is involved. At least I did for too long. Thank you!
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Yeah, with the law of sines, one can get such right angles, but this is basically by constructing a new set of triangles. See #743146
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