Question 138
Given $\triangle ABC$ and three points $D, E, F$ such that $DB = DC$, $EC = EA$, $FA = FB$, and $\angle^*BDC = \angle^*CEA = \angle^*AFB$ (here $\angle^*$ denotes a directed angle). Let $\Omega_D$ be the circle centered at $D$ passing through $B, C$, and define $\Omega_E$ and $\Omega_F$ similarly. Prove that the radical center of $\Omega_D, \Omega_E, \Omega_F$ lies on the Euler line of $\triangle DEF$.
Note: the Euler line of $\triangle DEF$ is the line through the orthocenter, centroid and circumcenter of $\triangle DEF$. The radical center of three circles is the point having equal power with respect to three circles in general position; that is, the common intersection point of the three pairwise radical axes.
Note: the Euler line of $\triangle DEF$ is the line through the orthocenter, centroid and circumcenter of $\triangle DEF$. The radical center of three circles is the point having equal power with respect to three circles in general position; that is, the common intersection point of the three pairwise radical axes.