22nd Putnam 1961

Problem B3

A, B, C, D lie in a plane. No three are collinear and the four points do not lie on a circle. Show that one point lies inside the circle through the other three.



If the convex hull is a triangle, then the fourth point lies inside its circumcircle. So suppose ABCD is convex. One pair of opposite angles must have sum greater than 180o (otherwise the points would lie on a circle). Suppose they are A and C. Then we claim that C lies inside the circle through A, B, D. Take a point C' on the ray DC on the far side of C from D such that ∠CBC' = ∠A + ∠C - 180o. Then ∠C' = 180o - ∠A. So C' lies on the circle. Hence C which lies on the segment C'D lies inside the circle.



22nd Putnam 1961

© John Scholes
15 Feb 2002