82. In a circle of radius $R$, two intersecting perpendicular chords are drawn.

a) Find the sum of the squares of the four segments of these chords, into which they are divided by the point of intersection.

b) Find the sum of the squares of the lengths of the chords, if the distance from the center of the circle to their point of intersection is $d$.
82. Let $AB$ and $CD$ be given chords, and $M$ be their point of intersection.

a) The arcs $\overline{AC}$ and $\overline{BD}$ together form a semicircle; therefore, $|AC|^{2} + |BD|^{2} = 4R^{2}$; thus,

$|AM|^{2} + |MC|^{2} + |MB|^{2} + |MD|^{2} = |AC|^{2} + |BD|^{2} = 4R^{2}$,

b) $|AB|^{2} + |CD|^{2} = (|AM| + |MB|)^{2} + (|CM| + |MD|)^{2} = |AM|^{2} + |MB|^{2} + |CM|^{2} + |MD|^{2} + 2|AM| \cdot |MB| + 2|CM| \cdot |MD| = 4R^{2} + 2(R^{2} - d^{2}) + 2(R^{2} - d^{2}) = 4(2R^{2} - d^{2})$.