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Miquel circles and a beautiful similarity

Source: IMO Shortlist 2006, Geometry 9, AIMO 2007, TST 2, P3

June 28, 2007
geometrycircumcircleIMO Shortlistgeometry solvedreflectionSpiral SimilarityMiquel point

Problem Statement

Points A1 A_{1}, B1 B_{1}, C1 C_{1} are chosen on the sides BC BC, CA CA, AB AB of a triangle ABC ABC respectively. The circumcircles of triangles AB1C1 AB_{1}C_{1}, BC1A1 BC_{1}A_{1}, CA1B1 CA_{1}B_{1} intersect the circumcircle of triangle ABC ABC again at points A2 A_{2}, B2 B_{2}, C2 C_{2} respectively (A2≠A,B2≠B,C2≠C A_{2}\neq A, B_{2}\neq B, C_{2}\neq C). Points A3 A_{3}, B3 B_{3}, C3 C_{3} are symmetric to A1 A_{1}, B1 B_{1}, C1 C_{1} with respect to the midpoints of the sides BC BC, CA CA, AB AB respectively. Prove that the triangles A2B2C2 A_{2}B_{2}C_{2} and A3B3C3 A_{3}B_{3}C_{3} are similar.
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