4. In $\triangle A B C$, $A B=A C=2$, and there are 100 different points $P_{1}, P_{2}, \cdots, P_{100}$ on side $B C$. Let $m_{i}=A P_{i}^{2}+$ $B P_{i} \cdot P_{i} C(i=1,2, \cdots, 100)$, find the value of $m_{1}+m_{2}+\cdots+m_{100}$.
(1990 National Junior High School League Problem)
4. Rotate $\triangle A B P_{i}$ counterclockwise around point $A$ to $\triangle A C P_{i}^{\prime}$, making $A B$ coincide with $A C$. Since $\angle A P_{i} C + \angle A P_{i}^{\prime} C = 180^{\circ}$, points $A$, $P_{i}$, $C$, and $P_{i}^{\prime}$ are concyclic. Let the intersection of $A C$ and $P_{i} P_{i}^{\prime}$ be point $D$. From $\triangle A P_{i} D \sim \triangle A C P_{i} \sim \triangle P_{i}^{\prime} C D$, we know $A P_{i}^{2} = A D \cdot A C$, $P_{i}^{\prime} C \cdot P_{i} C = D C \cdot A C$. Therefore, $m_{i} = A P_{i}^{2} + B P_{i} \cdot P_{i} C = (A D + D C) \cdot A C = A C^{2} = 4$. Hence, $m_{1} + m_{2} + \cdots + m_{100} = 400$.