Problem: (2007 Mock AIME 6 – #2) Draw in the diagonals of a regular octagon. What is the sum of all distinct angle measures, in degrees, formed by the intersections of the diagonals in the interior of the octagon?
Solution: Consider the circumscribed circle of the octagon. Each diagonal is a chord of this circle, and we know that the angle between two chords that intercept arcs of measure and is .
Now, any pair of diagonals can together intercept , , , , or little arcs (between vertices of the octagon). is not possible, because then on one side there is no little arc which means the intersection is not in the interior. This also means is not possible (they come in supplement pairs – except of course). Since each little arc is and we have to account for the division by , the final sum is
Comment: This wasn’t an extremely hard problem, but it needed some clever thinking. Most people overcounted, which is reasonable given that there are diagonals in an octagon. Pretty hard to draw accurately. Using arcs on a circle proved to be much easier and less prone to careless mistakes.
Practice Problem: Let be a set of points in the plane, no three of which lie on the same line. At most how many ordered triples of points in exist such that is obtuse?
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