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 4 of 5: Step 4

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AB^2+AC^2+BC^2=6R^2+2r^2

Detailed analysis

Using BC=2BM, AB^2=AP^2+(BM−PM)^2 and AC^2=AP^2+(BM+PM)^2, the sum is 2AP^2+6BM^2+2PM^2=6R^2+2r^2.