How do you find the area of the largest square?

Question

What formulas/theorems are to be used?

Answer 1

Note that $DF = EF=x$, i.e.

$$4\sin a + \frac 4{\cos a} = 3 \cos a + \frac3{\sin a}=x$$

which leads to $\tan a = \frac34$ and $x=\frac{37}5$. Thus, the area is

$$x^2 =\frac{1369}{25}$$