One of the things most pleasing to me about this whole inter-web-o-blog-o-sphere is that I can ask so many people for help in such a short amount of time. I am going to put the over-under on an answer at three hours (I would put it lower if I was on the West Coast) [Update: It turns out the winning time is 33 minutes. Thanks, Justin!].
In a 2008 MAA Mathematics Magazine “Proof Without Words” article, Sidney H. Kung gave the following proof of the tangent of the sum formula: The proof is transparent to me in all but one step. Why does ? I guess it must have something to do with the line segments being chords in the circle, since I haven’t used the circle anywhere in the proof, but I’m just missing something obvious or forgetting something from geometry. Help!