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Triangle and Hexagon

Source: 2013 AIME I Problem 12

March 15, 2013
geometrytrigonometryAMCAIMEnumber theoryrelatively primearea of a triangle

Problem Statement

Let △PQR\triangle PQR be a triangle with ∠P=75∘\angle P = 75^\circ and ∠Q=60∘\angle Q = 60^\circ. A regular hexagon ABCDEFABCDEF with side length 1 is drawn inside △PQR\triangle PQR so that side AB‾\overline{AB} lies on PQ‾\overline{PQ}, side CD‾\overline{CD} lies on QR‾\overline{QR}, and one of the remaining vertices lies on RP‾\overline{RP}. There are positive integers aa, bb, cc, and dd such that the area of △PQR\triangle PQR can be expressed in the form a+bcd\tfrac{a+b\sqrt c}d, where aa and dd are relatively prime and cc is not divisible by the square of any prime. Find a+b+c+da+b+c+d.
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