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2020 PUMaC Geometry A8

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December 31, 2021
geometry

Problem Statement

A1A2A3A4A_1A_2A_3A_4 is a cyclic quadrilateral inscribed in circle Ω\Omega, with side lengths A1A2=28A_1A_2 = 28, A2A3=123A_2A_3 =12\sqrt3, A3A4=283A_3A_4 = 28\sqrt3, and A4A1=8A_4A_1 = 8. Let XX be the intersection of A1A3,A2A4A_1A_3, A_2A_4. Now, for i=1,2,3,4i = 1, 2, 3, 4, let ωi\omega_i be the circle tangent to segmentsAiX A_iX, Ai+1XA_{i+1}X, and Ω\Omega, where we take indices cyclically (mod 44). Furthermore, for each ii, say ωi\omega_i is tangent to A1A3A_1A_3 at XiX_i , A2A4A_2A_4 at YiY_i , and Ω\Omega at TiT_i . Let P1P_1 be the intersection of T1X1T_1X_1 and T2X2T_2X_2, and P3P_3 the intersection of T3X3T_3X_3 and T4X4T_4X_4. Let P2P_2 be the intersection of T2Y2T_2Y_2 and T3Y3T_3Y_3, and P4P_4 the intersection of T1Y1T_1Y_1 and T4Y4T_4Y_4. Find the area of quadrilateral P1P2P3P4P_1P_2P_3P_4.
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