constructing an ellipse within a rhombus using a straight edge and compass Using a straight edge, compass and string, is it possible to construct an ellipse within any specified rhombus so that each side of the rhombus at their mid points is tangential to the ellipse?
 A: Yes.
An af-fine transformation converts the figure to a circle inscribed in a square. The major and minor axes of the ellipse then correspond to the diameters of the transformed circle that align with the diagonals of the square, and of course the midpoint of each diameter us the center of the curcle and the square.
Each diameter has a length equal to $1/\sqrt2$ times the diagonal in the circle-square figure. So the major and minor axis endpoints of the ellipse in the rhombus are between the center and vertices of the rhombus, with each axis having $1/\sqrt2$ times the length of its diagonal and each with its midpoint at the center of the rhombus.
You then know four points on the ellipse and the length of string required (the distance between the major axis vertices, $1/\sqrt2$ times the longer diagonal). With the major and minor axes known the foci are easily derived; the squared distance between the foci is the difference between the squared major and minor axes. With that you now have the placement and length of string both known and can draw the ellipse accordingly.
