MathDB

Equilateral triangle

T

is inscribed in circle

A

, which has radius

10

. Circle

B

with radius

3

is internally tangent to circle

A

at one vertex of

T

. Circles

C

and

D

, both with radius

2

, are internally tangent to circle

A

at the other two vertices of

T

. Circles

B

C

, and

D

are all externally tangent to circle

E

, which has radius

\frac {m}{n}

, where

m

and

n

are relatively prime positive integers. Find m \plus{} n. [asy]unitsize(2.2mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4;
pair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90); pair Ep=(0,4-27/5); pair[] dotted={A,B,C,D,Ep};
draw(Circle(A,10)); draw(Circle(B,3)); draw(Circle(C,2)); draw(Circle(D,2)); draw(Circle(Ep,27/5));
dot(dotted); label("

E

",Ep,E); label("

A

",A,W); label("

B

",B,W); label("

C

",C,W); label("

D

",D,E);[/asy]

Problem Statement