OME 2013 Problem 6

Let $ABCD$ a convex quadrilateral where: $|AB|+|CD|=\sqrt{2} |AC|$ and $|BC|+|DA|=\sqrt{2} |BD|$ What form does the quadrilateral have?

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.