AHSME 1962 Problem 15

Given triangle $ABC$ with base $AB$ fixed in length and position. As the vertex $C$ moves on a straight line, the intersection point of the three medians moves on: $\textbf{(A)}\ \text{a circle} \textbf{(B)}\ \text{a parabola} \textbf{(C)}\ \text{an ellipse} \textbf{(D)}\ \text{a straight line} \textbf{(E)}\ \text{a curve here not listed}$