MathDB

Area Ratio

Source:

December 6, 2005

geometryratioanalytic geometrytrigonometrynumber theoryrelatively prime

Problem Statement

In triangle

ABC

,

AB=13,

BC=15

and

CA=17.

Point

D

is on

\overline{AB},

E

is on

\overline{BC},

and

F

is on

CF=r\cdot CA,

Let

AD=p\cdot AB,

BE=q\cdot BC,

and

CF=r\cdot CA,

where

p,

q,

and

r

are positive and satisfy

p+q+r=2/3

and

p^2+q^2+r^2=2/5.

The ratio of the area of triangle

DEF

to the area of triangle

ABC

can be written in the form

m/n,

where

m

and

n

are relatively prime positive integers. Find

m+n.

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