ILL 1971 Problem 15

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 $$OV^2=\frac{d^2b^2(c + a)^2 - c^2a^2(d + b)^2}{ca(d - b)^2 - db(c - a)^2}.$$

Show also that if $\frac{c+a}{d+b}$ lies between $\frac{ca}{db}$ and $\sqrt{\frac{ca}{db}},$ and $\frac{c-a}{d-b}=\frac{ca}{db},$ then for a suitable choice of $\theta$ , a right circular cone exists with properties **(1)**and (2).