10. (22 points) In the polyhedron $A B C D E F$ shown in Figure 1, it is known that $A D, B E, C F$ are all perpendicular to the plane $A B C$.
Let $A D=a, B E=b, C F$ $=c, A B=A C=B C=$
1. Find the volume of the common part of 

tetrahedron $A B C E$ and 

tetrahedron $B D E F$ (expressed in terms of $a, b, c$).
10. As shown in Figure 2, let $A E$ intersect $B D$ and $B F$ intersect $C E$ at points $G$ and $H$, respectively. Then the tetrahedron $B E G H$ is the desired common part.
The calculations yield:
The distance from point $G$ to line $A B$ is $d_{1}=\frac{a b}{a+b}$, the distance from point $G$ to plane $B C F E$ is $d_{2}=\frac{\sqrt{3} d_{1}}{2 a}$, and the distance from point $H$ to line $B C$ is $d_{3}=\frac{b c}{b+c}$,
$$
S_{\triangle B E H}=\frac{b-d_{3}}{2} \text {. }
$$

Thus, $V_{\text {tetrahedron } B E G H}=\frac{S_{\triangle B E H} d_{2}}{3}=\frac{\sqrt{3} b^{3}}{12(a+b)(b+c)}$.