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Determine r1r2+r2r3+r1r3 - [IMO LongList 1971]

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January 1, 2011
geometry3D geometryprismtrigonometrygeometry unsolved

Problem Statement

The base of an inclined prism is a triangle ABCABC. The perpendicular projection of B1B_1, one of the top vertices, is the midpoint of BCBC. The dihedral angle between the lateral faces through BCBC and ABAB is α\alpha, and the lateral edges of the prism make an angle β\beta with the base. If r1,r2,r3r_1, r_2, r_3 are exradii of a perpendicular section of the prism, assuming that in ABC,cos⁡2A+cos⁡2B+cos⁡2C=1,∠A<∠B<∠C,ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1, \angle A < \angle B < \angle C, and BC=aBC = a, calculate r1r2+r1r3+r2r3.r_1r_2 + r_1r_3 + r_2r_3.
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