1967 IMO Problems/Problem 4
Let and be any two acute-angled triangles. Consider all triangles that are similar to (so that vertices , , correspond to vertices , , , respectively) and circumscribed about triangle (where lies on , on , and on ). Of all such possible triangles, determine the one with maximum area, and construct it.
We construct a point inside s.t. , where are a permutation of . Now construct the three circles . We obtain any of the triangles circumscribed to and similar to by selecting on , then taking , and then (a quick angle chase shows that are also colinear).
We now want to maximize . Clearly, always has the same shape (i.e. all triangles are similar), so we actually want to maximize . This happens when is the diameter of . Then , so will also be the diameter of . In the same way we show that is the diameter of , so everything is maximized, as we wanted.
This solution was posted and copyrighted by grobber. The thread can be found here: 
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