2018 AIME I Problem 15

David found four sticks of different lengths that can be used to form three noncongruent convex cyclic quadrilaterals, A, B, and 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.