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Circumferences tangents

Source: RMM 2016, Problem 5

February 28, 2016
geometryhexagonRMMRMM 2016

Problem Statement

A convex hexagon A1B1A2B2A3B3A_1B_1A_2B_2A_3B_3 it is inscribed in a circumference Ω\Omega with radius RR. The diagonals A1B2A_1B_2, A2B3A_2B_3, A3B1A_3B_1 are concurrent in XX. For each i=1,2,3i=1,2,3 let ωi\omega_i tangent to the segments XAiXA_i and XBiXB_i and tangent to the arc AiBiA_iB_i of Ω\Omega that does not contain the other vertices of the hexagon; let rir_i the radius of ωi\omega_i.
(a)(a) Prove that R≥r1+r2+r3R\geq r_1+r_2+r_3 (b)(b) If R=r1+r2+r3R= r_1+r_2+r_3, prove that the six points of tangency of the circumferences ωi\omega_i with the diagonals A1B2A_1B_2, A2B3A_2B_3, A3B1A_3B_1 are concyclic
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