Question 124
Let $ABC$ be an acute triangle. Points $D$, $E$ and $F$ lie on segments $BC$, $CA$ and $AB$ respectively, and each of the three segments $AD$, $BE$ and $CF$ contains the circumcenter of $ABC$. Prove that if any two of the ratios
$$
\frac{BD}{DC}, \frac{CE}{EA}, \frac{AF}{FB}, \frac{BF}{FA}, \frac{AE}{EC}, \frac{CD}{DB}
$$
are integers, then triangle $ABC$ is isosceles.
$$
\frac{BD}{DC}, \frac{CE}{EA}, \frac{AF}{FB}, \frac{BF}{FA}, \frac{AE}{EC}, \frac{CD}{DB}
$$
are integers, then triangle $ABC$ is isosceles.