2nd Swedish 1962 Problem 2 ABCD is a square side 1. P and Q lie on the side AB and R lies on the side CD. What are the possible values for the circumradius of PQR?
any value in [1/2, 1/√2] Solution Let O be the circumcenter. Then OP + OR ≥ PR ≥ AD = 1, so the radius is at least 1/2. P,Q,R always lie inside or on the circle through A,B,C,D which has radius 1/√2, so the radius is at most 1/√2. Now take ∠RPQ = 90 o , then QR is a diameter of the circumcircle and so the circumradius is ½QR. By adjusting the positions of Q, R we can obviously get QR to have any value in [1,√2]. 2nd Swedish 1962 © John Scholes jscholes@kalva.demon.co.uk 13 February 2004 Last corrected/updated 13 Feb 04