RIOPLATENSE 1996 Problem 4

Let $S$ be the circle of center $O$ and radius $R$ , and let $A, A'$ be two diametrically opposite points in $S$ . Let $P$ be the midpoint of $OA'$ and $\ell$ a line passing through $P$ , different from $AA '$ and from the perpendicular on $AA '$ . Let $B$ and $C$ be the intersection points of $\ell$ with $S$ and let $M$ be the midpoint of $BC$ . a) Let $H$ be the foot of the altitude from $A$ in the triangle $ABC$ . Let $D$ be the intersection point of the line $A'M$ with $AH$ . Determine the locus of point $D$ while $\ell$ varies . b) Line $AM$ intersects $OD$ at $I$ . Prove that $2 OI = ID$ and determine the locus of point $I$ while $\ell$ varies .