An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meters, respectively. What is the height, in meters, of the pillar at $E$? $\textbf{(A) }9 \qquad$ $\textbf{(B) } 6\sqrt{3} \qquad$ $\textbf{(C) } 8\sqrt{3} \qquad$ $\textbf{(D) } 17 \qquad$ $\textbf{(E) }12\sqrt{3}$