25th Brazil 2003 Problem B1 Given a circle and a point A inside the circle, but not at its center. Find points B, C, D on the circle which maximise the area of the quadrilateral ABCD.
If we fix the length BD, then we obviously maximise area BDA by taking the distance of A from BD as large as possible and hence by taking BD perpendicular to AO and on the opposite side of O to A. We maximise BCD by taking C the midpoint of the arc BD. So suppose the radius of the circle is R, OA = d and we take BD = 2x. We find area ABCD = x(R+d), which is maximised by maximising x. Thus we take BD to be the diameter perpendicular to AO and C the point of the circle on the ray AO. 25th Brazil 2003 © John Scholes jscholes@kalva.demon.co.uk 16 January 2004 Last corrected/updated 16 Jan 04