2014 AIME II Problems/Problem 14

14. In △ABC, AB=10, ∠A=30∘, and ∠C=45∘. Let H, D, and M be points on the line </math> such that $<math>\overline{AH}⊥$ (Error compiling LaTeX. Unknown error_msg)\overline{BC}$$ (Error compiling LaTeX. Unknown error_msg), $$ (Error compiling LaTeX. Unknown error_msg)∠BAD=∠CAD$$ (Error compiling LaTeX. Unknown error_msg), and $$ (Error compiling LaTeX. Unknown error_msg)BM=CM$$ (Error compiling LaTeX. Unknown error_msg). Point $$ (Error compiling LaTeX. Unknown error_msg)N$$ (Error compiling LaTeX. Unknown error_msg) is the midpoint of the segment $$ (Error compiling LaTeX. Unknown error_msg)\overline{HM}$$ (Error compiling LaTeX. Unknown error_msg), and point $$ (Error compiling LaTeX. Unknown error_msg)P$$ (Error compiling LaTeX. Unknown error_msg) is on ray AD such that $$ (Error compiling LaTeX. Unknown error_msg)\overline{PN}⊥\overline{BC}$$ (Error compiling LaTeX. Unknown error_msg). Then $$ (Error compiling LaTeX. Unknown error_msg)AP^2=\dfrac{m}{n}$</math>, where m and n are relatively prime positive integers. Find m+n.