83. Given two concentric circles with radii $r$ and $R (r < R)$. Through a point $P$ on the smaller circle, a line is drawn intersecting the larger circle at points $B$ and $C$. The perpendicular to $BC$ at point $P$ intersects the smaller circle at point $A$. Find $|PA|^2 + |PB|^2 + |PC|^2$.
83. If $M$ is the second intersection point of $B C$ with the smaller circle, then $|B M|=|P C|(M-$ between $B$ and $P), |B P|=|M P|+|B M|$, $|B P| \cdot|P C|=R^{2}-r^{2},\left.|M P+| P A\right|^{2}=4 r^{2}, \quad|P A|^{2}+|P B|^{2}+$ $+|P C|^{2}=|P A|^{2}+(|P B|-|P C|)^{2}+2|P B| \cdot|P C|=\left(|P A|^{2}+|M P|^{2}\right)+$ $+2(|P B| \cdot|P C|)=4 r^{2}+2\left(R^{2}-r^{2}\right)=2\left(R^{2}+r^{2}\right)$.