Question 123
On the three sides of triangle $ABC$, construct equilateral triangles $ABC_1$, $BCA_1$, $CAB_1$ outward (not overlapping $ABC$). Let $P$ be the second intersection point ($P \neq A$) of the circumcircles of $ABC_1$ and $CAB_1$. Let $Q$ be the point on the circumcircle of $CAB_1$ such that $PQ$ is parallel to $BA_1$, and let $R$ be the point on the circumcircle of $ABC_1$ such that $PR$ is parallel to $CA_1$.
Prove that the line joining the centroid of triangle $ABC$ and the centroid of triangle $PQR$ is parallel to the line $BC$.
Prove that the line joining the centroid of triangle $ABC$ and the centroid of triangle $PQR$ is parallel to the line $BC$.