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0413 geometry 4th edition Round 1 p3

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May 7, 2021
geometry4th edition

Problem Statement

Let Ω1(O1,r1)\Omega_1(O_1, r_1) and Ω2(O2,r2)\Omega_2(O_2, r_2) be two circles that intersect in two points X,YX, Y . Let A,CA, C be the points in which the line O1O2O_1O_2 cuts the circle Ω1\Omega_1, and let BB be the point in which the circle Ω2\Omega_2 itnersect the interior of the segment ACAC, and let MM be the intersection of the lines AXAX and BYBY . Prove that MM is the midpoint of the segment AXAX if and only if O1O2=12(r1+r2)O_1O_2 =\frac12 (r_1 + r_2).
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