1986 IMO Problems/Problem 4
Let be adjacent vertices of a regular -gon () with center . A triangle , which is congruent to and initially coincides with , moves in the plane in such a way that and each trace out the whole boundary of the polygon, with remaining inside the polygon. Find the locus of .
Let the vertex which is adjacent to . While moves from to , it is easy to see is cyclic. Thus lies on the bisector of . Moreover, is the intersection of a circle passing through (the circumcircle of ) and with a fixed radius (the radius is a function of ). Therefore varies in a line segment ended in . When and pass through the other sides, we get as locus distinct line segments, each passing throught and contained in (but not in ) for some vertex of the polygon. Each two of these lines are obtained one from another by a rotation with center .
This solution was posted and copyrighted by feliz. The original thread for this problem can be found here: 
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