Answer: 7

Small Hint:

The equal quantities are squared distances: with $D = (0,b),$ $E = (x,0),$ $F = (a,b-y)$ in an $a \times b$ rectangle, triangle $DEF$ is equilateral

Big Hint:

With $\theta = \angle ADE,$ equal sides force $\frac{a}{b} = \frac{\cos(30^\circ - \theta)} {\cos\theta},$ which increases in $\theta,$ while $y \ge 0$ caps $\theta$ at $30^\circ$

Solution:

Draw the rectangle with vertices $A = (0, 0),$ $B = (a, 0),$ $C = (a, b),$ $D = (0, b),$ and let $E = (x, 0)$ on $\overline{AB}$ and $F = (a, b - y)$ on $\overline{BC}.$ Then $DE^2 = b^2 + x^2,$ $DF^2 = a^2 + y^2,$ and $EF^2 = (a - x)^2 + (b - y)^2,$ so the system says exactly that triangle $DEF$ is equilateral, with the constraints keeping $E$ and $F$ on those two sides.

Let $\theta = \angle ADE,$ so $x = b\tan\theta$ and $DE = \frac{b}{\cos\theta}.$ Since $\angle EDF = 60^\circ$ and the corner angle at $D$ is $90^\circ,$ we get $\angle CDF = 30^\circ - \theta,$ so $y = a\tan(30^\circ - \theta)$ and $DF = \frac{a}{\cos(30^\circ - \theta)}.$ Setting $DE = DF$ gives $$\begin{aligned} \frac{a}{b} &= \frac{\cos(30^\circ - \theta)}{\cos\theta} \\ &= \cos 30^\circ + \sin 30^\circ \tan\theta, \end{aligned}$$ which is increasing in $\theta.$ The requirement $x \ge 0$ gives $\theta \ge 0,$ while $y \ge 0$ forces $\theta \le 30^\circ.$

The maximum is therefore at $\theta = 30^\circ,$ where $\frac{a}{b} = \frac{\sqrt{3}}{2} + \frac{1}{2\sqrt{3}} = \frac{2}{\sqrt{3}},$ attained with $y = 0$ and $x = \frac{b}{\sqrt{3}} \lt a.$ Hence $\rho^2 = \frac{4}{3},$ and $m + n = 4 + 3 = 7.$