geometry - A rectangle of a given aspect ratio inscribed in a hexagon.

I'm trying to find the largest rectangle of a given aspect ratio that can be inscribed in a hexagon.

I'm able to sort of walk through the problem in reverse, i.e. given an x, I can calculate the rectangle and aspect ratio :

1. find the point on AB for x


2. find the point on DE for x



3. find the point on BC across from #1


4. calculate height : distance between #1 & #2


5. calculate width : distance between #1 & #3


6. Aspect ratio is #5 / #4

How would I go about doing this in reverse? Thanks in advance for any guidance!