A dynamic view of this construction will be helpful in what follows.
or 

(1) 
Now consider the similar triangles TAG and TDH in the plane through O, G, and D, the axial plane. The triangles are similar because they each have a right angle and opposite vertical angles. This in turn implies
or 

(2) 
Also in the axial plane are the pairs of similar triangles HDT and IEA, and BDT and BEA. From these we see that
or 

and 
(3) 
. 
(4) 
or 

. 
(5) 
Notice also that in triangle IEA the line OL bisects AE so it must also bisect AI, making IA = 2AL. Putting this together with (1) and (2) we have
. 
(5) 
This might not look like an equation that we recognize, but if we let KT = y, the distance from the center of the ellipse to T be x, AB = 2a, and 2AL = p we have
.
This does look like the equation of an ellipse.