15 - 75 Triangle Redux

I came up with this problem after looking at the original one from @five_triangles (Find the area of the trapezoid ABCD) That’s a lot of fun but along the way while modelling the solution in geogebra I noticed AF is also on the diagonal of the trapezoid.

Note the 15-75-90 triangles at the bottom. Nowadays when I see them I also think of the following construction:  sometimes-one-direction-is-a-lot-tricker-than-the-other

which allows one to find the ratio of the sides without trigonometry:  1 : $2 - \sqrt{3}$ for the legs.

Bonus: Another problem with one of these in it: The $2 - \sqrt{3}$ in the expression is a dead giveaway.