Prove that the product of the diagonals is the sum of the product of the pairs of opposite sides

Proof:

1. Explain the construction
2. Prove that the triangles in the first pair are similar.
3. Prove that the triangles in the second pair are similar.
4. Write out the proportions deduced from the above steps
5. Finish the proof
(solution at the bottom of this page)

And what does this theorem say when the quadrilateral is a rectangle?

spoilers:

and therefore and therefore
adding both equations we get