Given the ellipse (e) with axes a , b, consider also its auxiliary circle (c). The polar lines f, g of a point A w.r. to (e) and (c) intersect at a point A* on the great axis of (e).
In fact, consider the intersection point A* of the polar f of A w.r. to the ellipse (e). By the discussion in CommonPolar.html The polar h of A* is the same w.r. to (e) as well as w.r. to (c) and passes through A. By the duality of pole-polar the polar g of A w.r. to (c) will pass also through A*.
Note in the above figure the various coincidences of the lines CF, DE, etc. with points of the polar h of A*. They are consequences of the previous property.