another yucky geometry question

[TJ Frame]

Hi everyone,
 Does anyone know a good place to look for ( or know of any example stacks)
that might shed light on how to plot an ellipse inscribed within a 4 sided
polygon like in the examples I created here:
http://www.tjframe.com/EllipseExample.htm
 I can handle the polygon, as well as finding the midpoints of the lines,
the center of the polygon etc. ..just not sure about fitting that ellipse
inside it. Curse my math - resistant brain!
 Any pointers would be, as usual, greatly appreciated :)
  - TJ