MathDB
Parallelograms and half-circles

Source: 2021 Mexico Center Zone Regional Olympiad, problem 5

January 17, 2022
Mexicogeometryparallelogrampower of a point

Problem Statement

Let ABCDABCD be a parallelogram. Half-circles ω1,ω2,ω3\omega_1,\omega_2,\omega_3 and ω4\omega_4 with diameters AB,BC,CDAB,BC,CD and DADA, respectively, are erected on the exterior of ABCDABCD. Line l1l_1 is parallel to BCBC and cuts ω1\omega_1 at XX, segment ABAB at PP, segment CDCD at RR and ω3\omega_3 at ZZ. Line l2l_2 is parallel to ABAB and cuts ω2\omega_2 at YY, segment BCBC at QQ, segment DADA at SS and ω4\omega_4 at WW. If XP⋅RZ=YQ⋅SWXP\cdot RZ=YQ\cdot SW, prove that PQRSPQRS is cyclic.
Proposed by José Alejandro Reyes González
Back to ProblemsView on AoPS