1988 AIME Problem 12

Question

Let

P

be an interior point of triangle

ABC

and extend lines from the vertices through

P

to the opposite sides. Let

a,

b,

c,

and

d

denote the lengths of the segments indicated in the figure. Find the product

abc

if

a+b+c=43

and

d=3.

Answer

Express the three barycentric coordinates of

P

using the ratios

a:d,

b:d,

c:d

Use

\frac3{a+3}+\frac3{b+3}+\frac3{c+3}=1

and expand symmetrically

Along the cevian from

A,

the barycentric coordinate at

A

is

\frac{d}{a+d},

and similarly at the other vertices. Since the three coordinates sum to

1

and

d=3,

\frac3{a+3}+\frac3{b+3}+\frac3{c+3}=1.

Put

s_1=a+b+c=43,

s_2=ab+bc+ca,

and

s_3=abc.

Clearing denominators gives

\begin{aligned}3(s_2+6s_1+27)&=s_3+3s_2\\&\quad+9s_1+27.\end{aligned}

Therefore

s_3=9s_1+54,

so

s_3=9(43)+54=441.

Problem 12 in other years