Let us consider 2 concentric circle radii R and r (R > r) with centre O.We fix P on the small circle and consider the variable chord PA of the small circle. Points B and C lie on the large circle; B,P,C are collinear and BC is perpendicular to AP.
i.) For which values of $\angle OPA$ do you think is the sum $AB^2+BC^2+CA^2$ extremal?
ii.) What are the possible positions of the midpoints U of BA and V of AC as varies?
Source; IMO 1988/Problem 1.
I was thinking about some solution involving coordinate geometry but I am interested in other synthetic geometry solutions as well.(In fact, am I right when I believe the first question is asking for the possible values of the given sum?) Thank you!