1. In $\triangle A B C$, $A B=A C=2, B C$ side has 100 different points $P_{1}, P_{2}, \cdots, P_{100}$, let $m_{i}=A P_{2}^{2}+$ $B P_{i} \cdot P_{i} C(i=1,2, \cdots, 100)$, find the value of $m_{1}+m_{2}+\cdots+m_{100}$.
1. Since $A B=A C$, by Corollary 1 of Stewart's Theorem, we have $A P_{i}^{2}=A B^{2}-B P_{i} \cdot P_{i} C$, thus $A P_{i}^{2}+B P_{i} \cdot P_{i} C=A B^{2}$, which means $m_{i}=A P_{i}^{2}+B P_{i} \cdot P_{i} C=A B^{2}=4$, hence $m_{1}+m_{2}+\cdots+m_{100}=4 \cdot 100=400$.