MathDB

Let

ABCD

be an isosceles trapezoid with

\overline{AD}\parallel{}\overline{BC}

whose angle at the longer base

\overline{AD}

\dfrac{\pi}{3}

. The diagonals have length

10\sqrt {21}

, and point

E

is at distances

10\sqrt {7}

and

30\sqrt {7}

from vertices

A

and

D

, respectively. Let

F

be the foot of the altitude from

C

\overline{AD}

. The distance

EF

can be expressed in the form

m\sqrt {n}

, where

m

and

n

are positive integers and

n

is not divisible by the square of any prime. Find m \plus{} n.

Problem Statement