MathDB

Let

ABCD

be a rectangle with

AB=10

and

BC=26

. Let

\omega_1

be the circle with diameter

\overline{AB}

and

\omega_2

be the circle with diameter

\overline{CD}

. Suppose

\ell

is a common internal tangent to

\omega_1

and

\omega_2

and that

\ell

intersects

AD

and

BC

E

and

F

respectively. What is

EF

?
[asy] size(10cm); draw((0,0)--(26,0)--(26,10)--(0,10)--cycle); draw((1,0)--(25,10)); draw(circle((0,5),5)); draw(circle((26,5),5)); dot((1,0)); dot((25,10)); label("

E

",(1,0),SE); label("

F

",(25,10),NW); label("

A

", (0,0), SW); label("

B

", (0,10), NW); label("

C

", (26,10), NE); label("

D

", (26,0), SE); dot((0,0)); dot((0,10)); dot((26,0)); dot((26,10)); [/asy]
Proposed by Nathan Xiong

Problem Statement