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Three circles concur

Source: Greece National Olympiad 2022, Problem 1

February 26, 2022
geometrycircumcircleconcurrency

Problem Statement

Let ABCABC be a triangle such that AB<AC<BCAB<AC<BC. Let D,ED,E be points on the segment BCBC such that BD=BABD=BA and CE=CACE=CA. If KK is the circumcenter of triangle ADEADE, FF is the intersection of lines AD,KCAD,KC and GG is the intersection of lines AE,KBAE,KB, then prove that the circumcircle of triangle KDEKDE (let it be c1c_1), the circle with center the point FF and radius FEFE (let it be c2c_2) and the circle with center GG and radius GDGD (let it be c3c_3) concur on a point which lies on the line AKAK.
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