Example 1. In $\triangle A B C$, $A B=A C=2, B C$ side has 100 different points $P_{1}, P_{2}, \cdots, P_{1} 00$. Let $m_{1}=A P_{1}^{2}+B P_{1} \cdot P_{1} C(i=1,2, \cdots, 100)$, then $m_{1}+m_{2}+\cdots+m_{100}$ equals what? (1990 National Junior High School Mathematics League)
Solve $\because A B$ $=A C$, according to Corollary 1 we have
$$
\begin{array}{c}
A P_{1}^{2}=A B^{2} \\
-B P_{\mathrm{i}} \cdot P_{1} C, \\
\therefore A P_{1}^{2} \\
+B P_{1} \cdot P_{1} C \\
=A B^{2}, \\
\therefore m_{\mathrm{i}}=A P_{1}^{2}+B P_{1} \cdot P_{\mathrm{i}} C \\
=A B^{2}=2^{2}=4,
\end{array}
$$

that is
$$
\begin{array}{c}
m_{1}+m_{2}+\cdots+m_{100} \\
=4 \times 100=400 .
\end{array}
$$