Ellipses are inscribed in a fixed convex quadrilateral with all sides of different lengths.
What is the locus of the center of the ellipses?
First I thought of an ellipse between a pair of parallel lines. Then the ellipse centers lay on a line halfway between the parallel lines.
In the case of a parallelogram there would be two different lines that the ellipse center must be on which means the locus of ellipse centers must be a single point at the center of the parallelogram.
Moving on to a trapezoid. There is still one center line that the ellipse centers must lay on. But not all points on the line are viable. Two degenerate cases occur when the ellipse degrades into the diagonals of the quadrilateral, which implies the centers of the diagonals are a limiting case.
I have not done much looking at quadrilaterals without parallel sides. But still, I conjecture that the locus of the ellipse centers is the line segment joining the centers of the diagonals of the quadrilateral.