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MA_1 is tangent to (A_1B_1C)

Source: French TST 2002

June 17, 2011
geometrycircumcirclegeometric transformationreflectiongeometry proposed

Problem Statement

In an acute-angled triangle ABCABC, A1A_1 and B1B_1 are the feet of the altitudes from AA and BB respectively, and MM is the midpoint of ABAB. a) Prove that MA1MA_1 is tangent to the circumcircle of triangle A1B1CA_1B_1C. b) Prove that the circumcircles of triangles A1B1C,BMA1A_1B_1C,BMA_1, and AMB1AMB_1 have a common point.
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