MathDB

Let

N_{14}

be the answer to problem 14.
Rectangle

ABCD

has area

\sqrt{2N_{14}}

. Points

E

F

G

, and

H

lie on the rays

\overrightarrow{AB}

\overrightarrow{BC}

\overrightarrow{CD}

, and

\overrightarrow{DA}

, respectively, such that

EFGH

is a rectangle with area

2\sqrt{2N_{14}}

that contains all of

ABCD

in its interior. If

\tan\angle AEH = \tan\angle BFE = \tan\angle CGF = \tan\angle DHG = \sqrt{\frac{1}{48}},

then

EG=\tfrac{m\sqrt{n}}{p}

, where

m

n

, and

p

are positive integers,

m

and

p

are relatively prime, and

n

is not divisible by the square of any prime. Compute

m + n + p

Problem Statement