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.

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M=\operatorname{mid}(PQ)\Longrightarrow M\text{ lies on the circle with diameter }OP

Detailed analysis

As Q moves on the smaller circle, M is its image under the homothety centered at P with factor 1/2. Hence M traces the circle with diameter OP.