Let ABC be a triangle with AB<AC<BC. On the side BC we consider points D
and E such that BA=BD and CE=CA. Let K be the circumcenter of triangle ADE and
let F, G be the points of intersection of the lines AD, KC and AE, KB respectively. Let ω1 be
the circumcircle of triangle KDE, ω2 the circle with center F and radius FE, and ω3 the circle
with center G and radius GD.
Prove that ω1, ω2, and ω3 pass through the same point and that this point of intersection lies on the line AK.
Balkanshortlist2021geometryconcurrency