2013 IMO Problems/Problem 3
Let the excircle of triangle opposite the vertex be tangent to the side at the point . Define the points on and on analogously, using the excircles opposite and , respectively. Suppose that the circumcentre of triangle lies on the circumcircle of triangle . Prove that triangle is right-angled.
Proposed by Alexander A. Polyansky, Russia
Let the excenters opposite be . Let the midpoint of be , which lies on , the nine-point circle of ; analogously define .
and , so (SAS), thus is equidistant from , with analogous results for . It follows that the circumcentre of is one of ; WLOG, suppose it is .
By isogonal conjugacy, concur at the Bevan point of . is the common perpendicular bisector of and , so . is the circle on diameter , so by Reim's theorem, .
Hence , as required.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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