If $A, B$, and $C$ are the tops of the heads of three successive people and $D, E$, and $F$ are their respective feet, let $P$ be the foot of the perpendicular from $A$ to $B E$ and let $Q$ be the foot of the perpendicular from $B$ to $C F$. Then, by equal angles, $\triangle A B P \sim \triangle B C Q$, so $$\frac{C F}{B E}=\frac{C F}{B Q}=\frac{C Q}{B Q}+1=\frac{B P}{A P}+1=\frac{B E}{A P}=\frac{B E}{A D}$$ Therefore the heights of successive people are in geometric progression. Hence, the heights of all five people are in geometric progression, so the middle height is $\sqrt{3 \cdot 7}=$ $\sqrt{21}$ feet.