MathDB

Circles

C_1

and

C_2

are externally tangent, and they are both internally tangent to circle

C_3

. The radii of

C_1

and

C_2

are

4

and

10

, respectively, and the centers of the three circles are all collinear. A chord of

C_3

is also a common external tangent of

C_1

and

C_2

. Given that the length of the chord is

\frac{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, find

m+n+p

Problem Statement