Here's an interesting problem, and result, that I wish to share with the math community here at Math SE. I think I've found a proof without words...
I came across this problem sometime back, and here's a possible proof without words that I'll post as a picture -
I claim that one can, for any inscribed quadrilateral EFGH, make necessary constructions (several parallelograms) as in the figure, and hence show that at least one of the diagonals is parallel to a side of the quadrilateral ABCD.
Is there anything I'm missing, or is the proof complete? Let me know!
Also, please post other proofs in the answers section! (So we can all solve the problem together and discuss several methods for the same - it'll be of use to everyone to know all possible methods of approaching this problem)
I was thinking about a possible solution using complex numbers, assuming one of the side pair to be parallel to the real axis in Argand's plane, for simplicity. A similar solution, not involving too much calculation can be done using vectors! I'm not sure about coordinate geometry, as it tends to get quite cumbersome in such situations, however if anyone does manage to prove it neatly, please post the solution.
Share your ideas!