Consider two concentric circles c1(O, a=OE) and c2(O, b=OF). Then a moving point J on the outer circle. Let G be the intersection point of the inner circle with the radius OJ. From J and G draw parallels to the sides of an angle w respectively. Then the intersection point K of these parallels moves along an ellipse (e).
In the case the angle w is a right one, this is a consequence of the discussion in Auxiliary.html . In that case the ratio JL/KL = a/b and the results of the reference apply.
In the other cases one can use the affinity representing the map of J to K. In fact, assuming that r = b/a one can calculate the coordinates of K(x',y') in terms of those of J(x,y) (taking the lines OL, ON as coordinate axes) and show that x' = x + ((1-r)/tan(w))*y, y' = r*y i.e the transformation is indeed an affinity. The affinity maps the circle c1 to the ellipse (e). Notice that (e) is contained in the strip between the tangents at circle c2 at the diametral points D, N.