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A difficult geometry problem

Source:

May 12, 2006
geometryratiotrigonometryarticlesAoPSwikigeometric transformationreflection

Problem Statement

In △ABC\triangle ABC, AB=360AB = 360, BC=507BC = 507, and CA=780CA = 780. Let MM be the midpoint of CA‾\overline{CA}, and let DD be the point on CA‾\overline{CA} such that BD‾\overline{BD} bisects angle ABCABC. Let FF be the point on BC‾\overline{BC} such that DF‾⊥BD‾\overline{DF} \perp \overline{BD}. Suppose that DF‾\overline{DF} meets BM‾\overline{BM} at EE. The ratio DE:EFDE: EF can be written in the form m/nm/n, where mm and nn are relatively prime positive integers. Find m+nm + n.
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