The diagram shows an equilateral triangle $ADE$ inside a square $ABCD$ . What is the value of $frac{[Delta ADE]}{[Delta DEC]}$ .

The diagram shows an equilateral triangle $ADE$ inside a square $ABCD$ . What is the value of $frac{[Delta ADE]}{[Delta DEC]}$ .

What I Tried: Here is the diagram :-

You can see I marked the angles as required. Now let $AB = x$ . We then have :-
$$[Delta ADE] = frac{sqrt{3}}{4}x^2$$
Now from here :- https://www.quora.com/What-is-the-ratio-of-sides-of-a-30-75-75-angle-triangle-without-sine-rule , I could understand and show that :- $$ED : DC : CE = bigg(frac{sqrt{3} + 1}{2} : frac{sqrt{3} + 1}{2} : 1bigg)$$

So let $EC = k$ , $CD = DE = frac{(sqrt{3} + 1)k}{2}$ .

From here :- $$x = frac{(sqrt{3} + 1)k}{2}$$
$$rightarrow k = EC = frac{2x}{(sqrt{3} + 1)}$$

Now, we can find area by Heron’s Formula. We have :- $$s = x + frac{x}{(sqrt{3} + 1)}$$
$$rightarrow s = frac{xsqrt{3} + 2x}{(sqrt{3} + 1)}$$
So :- $[Delta DEC] = sqrt{s(s-a)(s-b)(s-c)}$

$$rightarrow sqrt{Bigg(frac{(xsqrt{3} + 2x)}{(sqrt{3} + 1)}Bigg)Bigg(frac{x}{(sqrt{3} + 1)}Bigg)Bigg(frac{x}{(sqrt{3} + 1)}Bigg)Bigg(frac{(xsqrt{3})}{(sqrt{3} + 1)}Bigg)}$$

This looks like really a complicated expression, and I really am not going to attempt to simplify this. So can anyone give me a different solution?

Thank You.