ILL 1971 Problem 9

The base of an inclined prism is a triangle $ABC$ . The perpendicular projection of $B_1$ , one of the top vertices, is the midpoint of $BC$ . The dihedral angle between the lateral faces through $BC$ and $AB$ is $\alpha$ , and the lateral edges of the prism make an angle $\beta$ with the base. If $r_1, r_2, r_3$ are exradii of a perpendicular section of the prism, assuming that in $ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1, \angle A < \angle B < \angle C,$ and $BC = a$ , calculate $r_1r_2 + r_1r_3 + r_2r_3.$