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ASU 002 All Russian MO 1961 8.2 geometry

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June 17, 2019
geometrytangentialrectangle

Problem Statement

Given a rectangle A1A2A3A4A_1A_2A_3A_4. Four circles with AiA_i as their centres have their radiuses r1,r2,r3,r4r_1, r_2, r_3, r_4; and r1+r3=r2+r4<dr_1+r_3=r_2+r_4<d, where d is a diagonal of the rectangle. Two pairs of the outer common tangents to {the first and the third} and {the second and the fourth} circumferences make a quadrangle. Prove that you can inscribe a circle into that quadrangle.
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