Problem 1

Two coplanar circles with common center have radii

R>r

. Let P be fixed on the smaller circle and B vary on the larger circle. Line BP meets the larger circle again at C; the perpendicular to BP at P meets the smaller circle again at A (if tangent, A=P). (i) Find the set of values of

AB^2+BC^2+CA^2

. (ii) Find the locus of the midpoint of BC.

Step 2 of 5: Step 2

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AQ=2r,\quad AP^2=4r^2-4PM^2

Detailed analysis

AP is perpendicular to PQ at P, so angle APQ is right and AQ is a diameter of the smaller circle. Thus AQ=2r and AP^2=4r^2−4PM^2.