ISL 1988 Problem 71

The quadrilateral $A_1A_2A_3A_4$ is cyclic, and its sides are $a_1 = A_1A_2, a_2 = A_2A_3, a_3 = A_3A_4$ and $a_4 = A_4A_1.$ The respective circles with centres $I_i$ and radii $r_i$ are tangent externally to each side $a_i$ and to the sides $a_{i+1}$ and $a_{i-1}$ extended. ( $a_0 = a_4$ ). Show that $$\prod^4_{i=1} \frac{a_i}{r_i} = 4 \cdot (\csc (A_1) + \csc (A_2) )^2.$$