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Find a Length in a Rigid Triangle

Source: 2014 AIME II Problem 14

March 27, 2014
trigonometryAMCAIMEnumber theoryrelatively primegeometrytrig identities

Problem Statement

In △ABC\triangle ABC, AB=10AB=10, ∠A=30∘\angle A=30^\circ, and ∠C=45∘\angle C=45^\circ. Let H,DH,D, and MM be points on line BC‾\overline{BC} such that AH‾⊥BC‾\overline{AH}\perp\overline{BC}, ∠BAD=∠CAD\angle BAD=\angle CAD, and BM=CMBM=CM. Point NN is the midpoint of segment HM‾\overline{HM}, and point PP is on ray ADAD such that PN‾⊥BC‾\overline{PN}\perp\overline{BC}. Then AP2=mnAP^2=\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
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