(IMO 1988/1) Consider two circles of radii $R$ and $r$ $(R > r)$ with the same center. Let $P$ be a fixed point on the smaller circle and $B$ a variable point on the larger circle. The line $BP$ meets the larger circle again at $C$. The perpendicular $l$ to $BP$ at $P$ meets the smaller circle again at $A$. (As per our convention, if $l$ is tangent to the circle at $P,$ then we take $A = P$.)
(i) Find the set of values of $BC^2$ + $CA^2$ + $AB^2$
(ii) Find the locus of the midpoint of $AB$.
$X$ is midpoint of $AB$. Easy to see that $XC$ divides $OP$ in $1:2$ ratio. So a $-0.5$ homothety centered at midpoint of $OP$ sends $C$ to $X$. Since $C$ moves on a circle so $X$ moves on a circle too and the first part is obvious.