2001 AIME I Problems/Problem 9
In triangle , , and . Point is on , is on , and is on . Let , , and , where , , and are positive and satisfy and . The ratio of the area of triangle to the area of triangle can be written in the form , where and are relatively prime positive integers. Find .
We let denote area; then the desired value is
Using the formula for the area of a triangle , we find that
and similarly that and . Thus, we wish to find We know that , and also that . Substituting, the answer is , and .
By the barycentric area formula, our desired ratio is equal to so the answer is
Because the givens in the problem statement are all regarding the ratios of the sides, the side lengths of triangle , namely , are actually not necessary to solve the problem. This is clearly demonstrated in both of the above solutions, as the side lengths are not used at all.
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