Question 125
Let a triangle $ABC$ be given, and let $D$, $E$, $F$ be points on sides $BC$, $AC$, $AB$, respectively, such that
$$
\frac{BD}{CD} = \frac{CE}{AE} = \frac{AF}{BF}.
$$
Show that if the circumcenters of $ABC$ and $DEF$ coincide, then $ABC$ is equilateral.
$$
\frac{BD}{CD} = \frac{CE}{AE} = \frac{AF}{BF}.
$$
Show that if the circumcenters of $ABC$ and $DEF$ coincide, then $ABC$ is equilateral.