Question 2
Let $ABC$ be a triangle. Let $A'$ be the midpoint of the arc $BC$ of its circumcircle that does not contain $A$; define $B'$ and $C'$ similarly. The segments $A'B',B'C',C'A'$ meet the sides of $ABC$ at six points, two on each side, dividing each side into three parts. Prove that the three middle subsegments are equal if and only if $ABC$ is equilateral.