IMO 1971 LL GBR15

Origin: GBR

Problem

Let ABCD be a convex quadrilateral whose diagonals intersect at O at an angle \theta. Let us set OA = a, OB = b, OC = c, and OD = d, c > a > 0, and d > b > 0. Show that if there exists a right circular cone with vertex V , with the properties: (1) its axis passes through O, and (2) its curved surface passes through A, B, C and D, then OV 2 = d2b2(c + a)2 −c2a2(d + b)2 ca(d −b)2 −db(c −a)2 . 5 The numbers in the problem are not necessarily in base 10. Show also that if c+a d+b lies between ca db and  ca db, and c−a d−b = ca db, then for a suitable choice of \theta, a right circular cone exists with properties (1) and (2).