VMO 1975 Problem 3

Let $ABCD$ be a tetrahedron with $BA \perp AC,DB \perp (BAC)$ . Denote by $O$ the midpoint of $AB$ , and $K$ the foot of the perpendicular from $O$ to $DC$ . Suppose that $AC = BD$ . Prove that $\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}$ if and only if $2AC \cdot BD = AB^2$ .