The main diagonals AD, BE and CF of a (convex) cyclic hexagon ABCDEF are concurrent, if and only if, the two products of the alternate sides are equal: AB * CD * EF = BC * DE * FA.
BMO dual general
The result can be proved using ratios of similar triangles. A proof of the result and its converse is available on p. 99 of a useful book by Gardner, A.D. & Bradley, C.J. (2005). Plane Euclidean Geometry: Theory and Problems. University of Leeds, The United Kingdom Mathematics Trust (UKMT).
Unfortunately this condition is neither necessary nor sufficient for the concurrency of the main diagonals of a hexagon inscribed in a conic (as can easily be checked by the reader with dynamic geometry software). However, it might be an interesting exploration to try and find such a general condition?
Michael de Villiers, 30 October 2010.