Question 15
David found four sticks of different lengths that can be used to form three non-congruent 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 $m + n.$