How can I inscribe a rectangle or circle inside an arbitrary quadrilateral

Question

This may be a more math focused question, but wanted to ask here because it is in a CS context. I'm looking to inscribe a rectangle inside another (arbitrary) quad with the inscribed quad having the largest height and width possible. Since I think the algorithm will be similar, I'm looking to see if I can do this with a circle as well.

To be more clear hear is what I mean by the bounding quadrilateral as an example.

Here are 2 examples of the inscribed maximization I'm trying to achieve:

I have done some preliminary searching but have not found anything definitive. It seems that some form of dynamic programming could be the solution. It seems that this should be a linear optimization problem that should be more common than I have found, and perhaps I'm searching for the wrong terms.

Notes: For the inscribed square assume that we know a target w/h ratio we are looking for (e.g. 4:3). For the quad, assume that the sides will not cross and that it will be concave (if that simplifies the calculation).