David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, (A), (B), (C), which can each be inscribed in a circle with radius (1). Let (\\varphi_A) denote the measure of the acute angle made by the diagonals of quadrilateral (A), and define (\\varphi_B) and (\\varphi_C) similarly. Suppose that (\\sin\\varphi_A=\\frac{2}{3}), (\\sin\\varphi_B=\\frac{3}{5}), and (\\sin\\varphi_C=\\frac{6}{7}). All three quadrilaterals have the same area (K), which can be written in the form (\\frac{m}{n}), where (m) and (n) are relatively prime positive integers. Find (m+n).
Leading zeroes must be inputted, so if your answer is 34, then input 034. Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.