2004 AIME II Problems/Problem 13
Let the intersection of and be . Since it follows that is a parallelogram, and so . Also, as , it follows that .
By the Law of Cosines, . Thus the length similarity ratio between and is .
Let and be the lengths of the altitudes in to respectively. Then, the ratio of the areas .
However, , with all three heights oriented in the same direction. Since , it follows that , and from the similarity ratio, . Hence , and the ratio of the areas is . The answer is .
Additional Trigonometry-Free Alternative
Instead of using the Law of Cosines, we can draw a line perpendicular to line BC down from point A until it intersects BC at a point . Since , we can use the triangle to deduce that , and . From here, we can use Pythagorean theorem to deduce that . Then, we can follow with the rest of the solution above.
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