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triangles AEM , MCA are similar iff <ABC = 60^o

Source: Mexican Mathematical Olympiad 2006 P2

July 31, 2018
geometrysimilar triangles

Problem Statement

Let ABCABC be a right triangle with a right angle at AA, such that AB<ACAB < AC. Let MM be the midpoint of BCBC and DD the intersection of ACAC with the perpendicular on BCBC passing through MM. Let EE be the intersection of the parallel to ACAC that passes through MM, with the perpendicular on BDBD passing through BB. Show that the triangles AEMAEM and MCAMCA are similar if and only if ∠ABC=60o\angle ABC = 60^o.
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