2001 AIME I Problems/Problem 4
In triangle , angles and measure degrees and degrees, respectively. The bisector of angle intersects at , and . The area of triangle can be written in the form , where , , and are positive integers, and is not divisible by the square of any prime. Find .
and the answer is .
Since has a measure of , and thus has sines and cosines that are easy to compute, we attempt to find and , and use the formula that
By angle chasing, we find that is a triangle with and . Thus .
Switching to the lower triangle , , and , with .
Using the Law of Sines on :
We now plug in , and into the formula for the area:
Thus the answer is
Note: We could also get the lengths (and area) of the triangle by drawing a perpendicular from to , forming a and a triangle.
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