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(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 $AB$.

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    See that website: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=361291&sid=d23f4b5a68baa8584f932fa000074c31#p3612912012-06-19

2 Answers 2