Are there versions of the Intersecting chords/secants theorem that work on non-cyclic quadrilaterals?

Question

The Intersecting chords theorem

The Intersecting secants theorem

You know how the Cosine rule is a generalisation of Pythagoras' theorem in that it works for all triangles- not just right-angled ones.

I was wondering if there is a generalisation of either (or both together) of the above theorems that works for all quadrilaterals $ABCD,$ not just cyclic ones.