Question 137
Let $ABC$ be a triangle with $AB \lt AC \lt BC$. On the ray $BC$ we consider a point $D$ such that $BD = BA$ and on the ray $CB$ we consider a point $E$ such that $CE = CA$. Let $K$ be the circumcenter of the triangle $ADE$ and let $F$, $G$ be the intersections of the lines $AD$, $KC$ and $AE$, $KB$, respectively. Prove that the circumcircle of the triangle $KDE$, say $c_1$, the circle with center $F$ and radius $FE$, say $c_2$, and the circle with center $G$ and radius $GD$, say $c_3$, pass through a point on the line $AK$.