2
$\begingroup$

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!

  • 1
    No, question 1 is asking for the values the angle OPA, such that the sum is maximum or minimum.2011-10-21
  • 0
    Thank you.In fact,I tried connecting A, B and C to O.With O(0,0) as the origin, I assigned coordinates to A, B and C and obtained relations of the type $p^2+q^2=R^2$ and so on.I tried using the Pythagoras theorem and used the previous results of the type I mentioned,but I always seemed to get stuck while trying to obtain the sum mentioned in the problem as expression independent of the co-ordinates I had specified.Your insight is appreciated!2011-10-21
  • 0
    @SabyasachiMukherjee Are you still looking for solution, I can write one up for you.2012-04-12

1 Answers 1