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Outer tangents form a circumscribed quadrangle

Source: 1961 All-Soviet Union Olympiad

August 4, 2015
geometryrectanglecircumscribed quadrilateral

Problem Statement

Consider a rectangle A1A2A3A4A_1A_2A_3A_4 and a circle Ci\mathcal{C}_i centered at AiA_i with radius rir_i for i=1,2,3,4i=1,2,3,4. Suppose that r1+r3=r2+r4<dr_1+r_3=r_2+r_4<d, where dd is the diagonal of the rectangle. The two pairs of common outer tangents of C1\mathcal{C}_1 and C3\mathcal{C}_3, and of C2\mathcal{C}_2 and C4\mathcal{C}_4 form a quadrangle. Prove that this quadrangle has an inscribed circle.
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