MathDB

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

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