1998 AIME Problems/Problem 12
From AoPSWiki
Problem
Let 
be equilateral, and 
and 
be the midpoints of 
and 
respectively. There exist points 
and 
on 
and 
respectively, with the property that 
is on 
is on 
and is on 
The ratio of the area of triangle to the area of triangle 
is 
where 
and 
are integers, and is not divisible by the square of any prime. What is 
?
Solution
We let 
, 
, 
. Since 
and 
, 
and 
.
By alternate interior angles, we have 
and 
. By vertical angles, 
.
Thus 
, so 
.
Since 
is equilateral, 
. Solving for 
and 
using 
and 
gives 
and 
.
Using the Law of Cosines, we get
We want the ratio of the squares of the sides, so 
so 
.
See also
| 1998 AIME (Problems • Resources) | ||
| Preceded by Problem 11 | Followed by Problem 13 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
