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BM=CF and <MBC=<FCA means ABC is equilateral

Source: Pan African Olympiad 2010

October 1, 2011
geometrycircumcirclegeometry proposed

Problem Statement

In an acute-angled triangle ABCABC, CFCF is an altitude, with FF on ABAB, and BMBM is a median, with MM on CACA. Given that BM=CFBM=CF and ∠MBC=∠FCA\angle MBC=\angle FCA, prove that triangle ABCABC is equilateral.
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