2005 AIME I Problems/Problem 15
Problem
Triangle 
has 
The incircle of the triangle evenly trisects the median 
If the area of the triangle is 
where 
and 
are integers and is not divisible by the square of a prime, find 
Solution
Let 
and 
be the points of tangency of the incircle with 
and 
, respectively. Without loss of generality, let 
, so that is between 
and 
. Let the length of the median be 
. Then by two applications of the Power of a Point Theorem, 
, so 
. Now, 
and 
are two tangents to a circle from the same point, so 
and thus 
. Then triangle 
is isosceles,
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