Let $\Delta_1$ and $\Delta_2$ be two triangles in a plane with centroids $G_1$ and $G_2$ respectively. Let $X$, $Y$ be variable points on the perimeter of the triangles $\Delta_1$,$\Delta_2$ respectively. Suppose $M(X,Y)$ is the midpoint of the segment $XY$. Describe the locus of $M(X,Y)$.
I have absolutely no idea how to approach this problem. Hints are preferred to full answers.
Hint: A triangle is the convex envelope of its vertices $A,B,C$, i.e. the set of linear combinations $$ \alpha A+\beta B+\gamma C $$ with $\alpha,\beta,\gamma\in[0,1]$ and $\alpha+\beta+\gamma=1$. The boundary is given by the set of the previous linear combinations in which $\alpha,\beta$ or $\gamma$ equals $0$. If we have two triangles, our locus is given by the set of linear combinations $$ \frac{\alpha}{2}A+\frac{\beta}{2}B+\frac{\gamma}{2}C+\frac{\alpha'}{2}A'+\frac{\beta'}{2}B'+\frac{\gamma'}{2}C' $$ such that $\alpha+\beta+\gamma=\alpha'+\beta'+\gamma'=1$, $\alpha,\beta,\gamma,\alpha',\beta',\gamma'\in[0,1]$ and $\alpha\beta\gamma=\alpha'\beta'\gamma'=0$.
That is a subset of the boundary of the convex envelope of $A,B,C,A',B',C'$.