ILL 1990 Problem 70

$BC$ is a segment, $M$ is point on $BC$ , $A$ is a point such that $A, B, C$ are non-collinear.

(i) Prove that if $M$ is the midpoint of $BC$ , then $AB^2 + AC^2 = 2(AM^2 + BM^2).$ (ii) If there exists another point (except $M$ ) on segment $BC$ satisfying (i), find the region of point $A$ might occupy.