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BMO Shortlist 2021 G1

Source: BMO Shortlist 2021

May 8, 2022
Balkanshortlist2021geometryconcurrency

Problem Statement

Let ABCABC be a triangle with AB<AC<BCAB < AC < BC. On the side BCBC we consider points DD and EE such that BA=BDBA = BD and CE=CACE = CA. Let KK be the circumcenter of triangle ADEADE and let FF, GG be the points of intersection of the lines ADAD, KCKC and AEAE, KBKB respectively. Let ω1\omega_1 be the circumcircle of triangle KDEKDE, ω2\omega_2 the circle with center FF and radius FEFE, and ω3\omega_3 the circle with center GG and radius GDGD. Prove that ω1\omega_1, ω2\omega_2, and ω3\omega_3 pass through the same point and that this point of intersection lies on the line AKAK.
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