MathNet 2014 Question 46

Let $ABC$ be an acute triangle. Let $AM$, $BN$ and $CL$ be its medians, meeting at the centroid $G$. Let $M'$, $N'$ and $L'$ be the midpoints of $AG$, $BG$ and $CG$, respectively. Show that the six points $M,M',N,N',L,L'$ lie on a circle if and only if $ABC$ is equilateral.

Supplementary illustration (one configuration):