Construct focii of ellipse given center and four tangent lines We are given 4 distinct lines $a,b,c$ and $d$ which are said to be tangents to an ellipse. Let's consider that the 4 meetings of these lines form an convex quadrilateral $ABCD.$
There is a theorem which garantees that the center $E$ of ellipse inscribed in $ABCD$ must lie in the line joining the midpoints of $AC$ and $BD$ (diagonals of $ABCD).$
I've been wondering how can we construct such an ellipse (namely the focii or the axis of the ellipse) with the four lines and its center $E.$

EDIT: in case of people wondering: yes, any convex quad admits at least one (but usually infinite) inscribed ellipse
EDIT2: I forgot to mention that I'm looking for an straightedge and compass construction. Sorry.

To construct one of the ellipses inscribed in the given convex quadrilateral, let us use Brianchon's hexagram theorem (see the book cited bellow, pp. 261-265). I used Corollary 1 (p.263) in my construction:

An auxiliary green line cuts the quadrilateral to give a pentagone. The ellipse inscribed in this pentagone is unique. We then construct one point of the ellipse on each side. GeoGebra finishes the work.
To check the construction, have a look here. It is possible to move the big blue points.

The orange lines are used for construction of the center by hand.
Doerrie, Heinrich, 100 great problems of elementary mathematics. their history and solution. Transl. from the German by David Antin, New York: Dover Publications, Inc. X, 393 p. (1965). 
