Locus of the centroid

With an obvious notation the position vector of the centroid $G$ is $\begin{aligned} \mathbf{g} & = \; \tfrac13(\mathbf{d} + \mathbf{e} + \mathbf{f}) \\ \mathbf{g} - \tfrac12(\mathbf{e}+\mathbf{f}) & = \; \tfrac13\left[\mathbf{d} - \tfrac12(\mathbf{e}+\mathbf{f})\right] \end{aligned}$ so that the homothecy $H(M,\tfrac13)$ (enlargement with centre $M$ and scale factor $\tfrac13$ ), where $M$ is the midpoint of $EF$ , maps $D$ to $G$ . Thus the locus of $G$ is the image of the triangle $ABC$ under the homothecy $H(M,\tfrac13)$ , and hence is a triangle similar to $ABC$ , but of one third the linear dimension. Thus we deduce that $L = \tfrac19T$ , and hence $\tfrac{L}{T} = \tfrac19$ , making the answer $1+9=\boxed{10}$ .

G is the average of 3 points. Assuming E and F are fixed, any movement of D will cause G to move by $\frac13$ the distance in the same direction, so it describes a triangle that is $\frac13$ the size of $\triangle{ABC}$ in two dimensions, resulting in $\frac19$ its area.