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Convex Quadrilateral - [IMO LongList 1971]

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January 1, 2011
geometry3D geometrygeometry unsolved

Problem Statement

Let ABCDABCD be a convex quadrilateral whose diagonals intersect at OO at an angle θ\theta. Let us set OA=a,OB=b,OC=cOA = a, OB = b, OC = c, and OD=d,c>a>0OD = d, c > a > 0, and d>b>0.d > b > 0.
Show that if there exists a right circular cone with vertex VV, with the properties:
(1) its axis passes through OO, and
(2) its curved surface passes through A,B,CA,B,C and D,D, then OV2=d2b2(c+a)2−c2a2(d+b)2ca(d−b)2−db(c−a)2.OV^2=\frac{d^2b^2(c + a)^2 - c^2a^2(d + b)^2}{ca(d - b)^2 - db(c - a)^2}.
Show also that if c+ad+b\frac{c+a}{d+b} lies between cadb\frac{ca}{db} and cadb,\sqrt{\frac{ca}{db}}, and c−ad−b=cadb,\frac{c-a}{d-b}=\frac{ca}{db}, then for a suitable choice of θ\theta, a right circular cone exists with properties (1) and (2).
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