# 2014 AIME II Problems/Problem 14

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14. In △ABC, AB=10, ∠A=30∘, and ∠C=45∘. Let H, D, and M be points on the line $$\overline{BC}$$ such that \$$\overline{AH}⊥ (Error compiling LaTeX. ! Package inputenc Error: Unicode char \u8:⊥ not set up for use with LaTeX.)\overline{BC} (Error compiling LaTeX. ! Missing  inserted.),  (Error compiling LaTeX. ! Missing  inserted.)∠BAD=∠CAD (Error compiling LaTeX. ! Missing  inserted.), and  (Error compiling LaTeX. ! Missing  inserted.)BM=CM (Error compiling LaTeX. ! Missing  inserted.). Point  (Error compiling LaTeX. ! Missing  inserted.)N (Error compiling LaTeX. ! Missing  inserted.) is the midpoint of the segment  (Error compiling LaTeX. ! Missing  inserted.)\overline{HM} (Error compiling LaTeX. ! Missing  inserted.), and point  (Error compiling LaTeX. ! Missing  inserted.)P (Error compiling LaTeX. ! Missing  inserted.) is on ray AD such that  (Error compiling LaTeX. ! Missing  inserted.)\overline{PN}⊥\overline{BC} (Error compiling LaTeX. ! Missing  inserted.). Then  (Error compiling LaTeX. ! Missing  inserted.)AP^2=\dfrac{m}{n}$, where m and n are relatively prime positive integers. Find m+n.