Vertical pillars stand at the vertices of a regular hexagon $ABCDEF$ lying horizontally on the ground, holding up a flat (non-horizontal) solar panel. The pillars at $A,B,C$ have heights $14,11,12$ meters respectively. What is the height, in meters, of the pillar at $E$?

1. Let $O$ be the center of the hexagon. In the ground plane $ABCO$ is a rhombus, so its diagonals $AC$ and $BO$ have the same midpoint. The panel is flat, so the heights above these two diagonals have the same average: $h(A)+h(C)=h(B)+h(O)$.
2. Hence $h(O)=14+12-11=15$.
3. $O$ is the midpoint of $\overline{BE}$, so on the flat panel $h(O)=\frac{h(B)+h(E)}2$, i.e. $15=\frac{11+h(E)}2$.
4. $h(E)=\boxed{19}$.