1. Let the altitudes $A D, B E, C F$ of the acute triangle $\triangle A B C$ intersect at $H$. If $B C=a, C A=b, A B=c$, then the value of $A H \cdot$ $A D+B H \cdot B E+C H \cdot C F$ is ( ).
A. $\frac{a b+b c+c a}{2}$
B. $\frac{a^{2}+b^{2}+c^{2}}{2}$
C. $\frac{2(a b+b c+c a)}{3}$
D. $\frac{2\left(a^{2}+b^{2}+c^{2}\right)}{3}$
1.B. From $D, H, E, C$ being concyclic, we get $A H \cdot A D=A E \cdot A C=A C \cdot A B \cos \angle A=\frac{b^{2}+c^{2}-a^{2}}{2}$. Similarly, there are two other equations. Adding them up yields the result.