Question 69
An equilateral triangle $DEF$ is inscribed in triangle $ABC$, with $D,E,F$ on $BC,CA,AB$, respectively. Let $D_1,D_2,D_3$ be the midpoints of $BD,DC,BC$; let $E_1,E_2,E_3$ be the midpoints of $CE,EA,CA$; and let $F_1,F_2,F_3$ be the midpoints of $AF,FB,AB$. Choose points $P,Q,R$ outside $ABC$ so that $PD_1D_2,QE_1E_2,RF_1F_2$ are equilateral. Let $M_1,M_2,M_3$ be the centroids of $PDD_3,QEE_3,RFF_3$. Prove that $M_1M_2M_3$ is equilateral.
