What's the Altitude?

Let the side length of square $ABCD$ be $a$ . Then $\tan \theta = \dfrac {64}a$ and

$\begin{aligned} \tan (90^\circ - 2\theta) & = \cot 2\theta = \frac 1 {\tan 2\theta} = \frac {36}a \\ \frac {1-\tan^2 \theta}{2 \tan \theta} & = \frac {36}a \\ \frac {1-\left(\frac {64}a\right)^2}{2 \times \frac {64}a} & = \frac {36}a \\ 1 - \frac {64^2}{a^2} & = \frac {2 \times 64\times 36}{a^2} \\ \implies a^2 & = 8704 \\ a & = 16\sqrt{34} \end{aligned}$

The area of $\triangle AEF$ :

$\begin{aligned} [AEF] & = a^2 - \left(\frac {64a}2 + \frac {36a}2 + \frac {(a-64)(a-36)}2 \right) \\ & = a^2 - \left(\frac {a^2}2 + 1152\right) \\ & = \frac {a^2}2 - 1152 = \frac {8704}2 - 1152 = \boxed{3200} \end{aligned}$

The length of $EF$ is $\sqrt{(a-64)^2 + (a-36)^2} = 20\sqrt{57-8\sqrt{34}}$ and the altitude of $\triangle AEF$ , $d = \dfrac {3200 \times 2}{20\sqrt{57-8\sqrt{34}}} = \dfrac {320}{\sqrt{57-8\sqrt{34}}}$ . Therefore, $a+b+c + d* = 320 + 57 + 8 + 34 = \boxed{419}$ .

Area of $\triangle {AEF}$ is $3200$ . Length of side $\overline {EF}$ is $\sqrt {(36-\sqrt {8704})^2+(\sqrt {8704}-64)^2}=20\sqrt {57-8\sqrt {34}}$ . So the altitude is given by $d=\dfrac{320}{\sqrt {57-8\sqrt {34}}}$ . Therefore, $a=320, b=57, c=8, d=34$ and $a+b+c+d=\boxed {419}$

In square $ABCD$ let $AB = x \implies$

$AE = \sqrt{x^2 + 36^2}$ , $AF = \sqrt{x^2 + 64^2}$ and $\sin(2\theta) = \dfrac{x}{\sqrt{x^2 + 36^2}} = 2\sin(\theta)\cos(\theta)$ , where $\sin(\theta) = \dfrac{64}{\sqrt{x^2 + 64^2}}$

and $\cos(\theta) = \dfrac{x}{\sqrt{x^2 + 64^2}}$

$\implies \sin(2\theta) = \dfrac{x}{\sqrt{x^2 = 36^2}} = 2\sin(\theta)\cos(\theta) = \dfrac{128x}{x^2 + 64^2}$

$\implies 128\sqrt{x^2 + 36^2} = x^2 + 64^2 \implies x^4 - 2 * 64^2 x^2 - 64^2 * 8^2 * 17 = 0$

$\implies x^2 = 4096 \pm 4608$

Choosing the positive real root $\implies x = \sqrt{8704} = 16\sqrt{34}$

To find the distance $d$ :

$\vec{U} = (\sqrt{8704} - 36)\vec{i} + (64 - \sqrt{8704})\vec{j} + 0\vec{k}$

$\vec{V} = \sqrt{8704} \:\ \vec{i} + 64\vec{j} + 0\vec{k}$

$\implies |\vec{U} X \vec{V}| = |64\sqrt{8704} - 2304 - 64\sqrt{8704} + 8704| = 6400$

and $\vec{U} = 4\sqrt{(4\sqrt{34} - 9)^2 + (16 - 4\sqrt{34})^2} = 20\sqrt{57 - 8\sqrt{34}}$

$\implies d = \dfrac{320}{\sqrt{57 - 8\sqrt{34}}} = \dfrac{a}{\sqrt{b - c\sqrt{d^{*}}}} \implies$ $a + b + c + d^{*} = \boxed{419}$ .