2006 AIME II Problems/Problem 12
Equilateral is inscribed in a circle of radius . Extend through to point so that and extend through to point so that Through draw a line parallel to and through draw a line parallel to Let be the intersection of and Let be the point on the circle that is collinear with and and distinct from Given that the area of can be expressed in the form where and are positive integers, and are relatively prime, and is not divisible by the square of any prime, find
Notice that because . Also, because they both correspond to arc . So .
Because the ratio of the area of two similar figures is the square of the ratio of the corresponding sides, . Therefore, the answer is .
Solution 2: Analytic Geometry
Solution by e_power_pi_times_i
Let the center of the circle be and the origin. Then, , , . and can be calculated easily knowing and , , . As and are parallel to and , . and is the intersection between and circle . Therefore . Using the Shoelace Theorem, , so the answer is
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