MathDB

Congruent Incircles

Source: AIME 2010I Problem 15

March 17, 2010

geometryratioincentergeometric transformationquadraticstrigonometryhomothety

Problem Statement

\triangle{ABC}

with

AB = 12

BC = 13

, and

AC = 15

, let

M

be a point on

\overline{AC}

such that the incircles of

\triangle{ABM}

and

\triangle{BCM}

have equal radii. Let

p

and

q

be positive relatively prime integers such that

\tfrac{AM}{CM} = \tfrac{p}{q}

. Find

p + q