I am trying to rigorously prove the POP theorem. Here is perhaps the best one available: http://www.cut-the-knot.org/pythagoras/PPower.shtml Can someone link me to a more complete proof? A good proof would need to cover all of the cases: when P is inside the circle, on the circle (a rather obvious case), or outside the circle, and when the lines are tangents, secants, or chords. Also, I think that link uses a geometric theorem that goes something like angles intercepting the same arc are equal, which I am also having difficulty proving. So, if someone could link me to a proof of that or provide a short one here that would be great.