ellipse inscribed in quadrilateral
this work show how to inscribe an ellipse in a generic quadrilateral. you can move the point of tangency "G", as well as "A" "B" "D". "C" can be moved in gray region. when "C" is over the border of gray region another similar work is required. then points "E" "F" go to infinity and some lines are parallel. if you need another generic quadrilateral with "C" out of gray region you must reorder your vertexes.