MathDB

Tangent Circles

Source: AIME II 2007 #15

March 29, 2007

geometryinradiusincentergeometric transformationhomothetycircumcircleratio

Problem Statement

Four circles

\omega,

\omega_{A},

\omega_{B},

and

\omega_{C}

with the same radius are drawn in the interior of triangle

ABC

such that

\omega_{A}

is tangent to sides

AB

and

AC

,

\omega_{B}

to

BC

and

BA

,

\omega_{C}

to

CA

and

CB

, and

\omega

is externally tangent to

\omega_{A},

\omega_{B},

and

\omega_{C}

. If the sides of triangle

ABC

are

13,

14,

and

15,

the radius of

\omega

can be represented in the form

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. Find

m+n.

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