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 3 of 5: Step 3

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BM^2=R^2-OM^2=R^2-r^2+PM^2

Detailed analysis

The large-circle radius relation gives BM^2=R^2−OM^2. Since OM^2=r^2−PM^2, BM^2=R^2−r^2+PM^2.