The foci of an ellipse follow readily if one has the rectangle formed by four intersections with a concentric circle. To construct the center of an ellipse, inscribe an arbitrary pentagram (green cyan, yellow,magenta, blue), and use Pascal's theorem to construct (without a compass!) tangents (black) at three of the vertices. Bisect two sides of the pentagram joining points of tangency. The medians (cyan, yellow) from the intersections of pairs of tangents to these midpoints go on to intersect at the ellipse's center. (Click to enlarge.)