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Equilaterals outside ABC

Source: Argentina IMO 2005 TST, problem 3

April 28, 2005
ratiogeometrygeometric transformationrotationpower of a pointradical axisgeometry proposed

Problem Statement

Given the triangle ABCABC we consider the points X,Y,ZX,Y,Z such that the triangles ABZ,BCX,CAZABZ,BCX,CAZ are equilateral, and they don't have intersection with ABCABC. Let B′B' be the midpoint of BCBC, N′N' the midpoint of CYCY, and M,NM,N the midpoints of AZ,CXAZ,CX, respectively. Prove that B′N′⊥MNB'N' \bot MN.
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