2012 AIME II Problems/Problem 15
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Problem 15
Triangle is inscribed in circle with , , and . The bisector of angle meets side at and circle at a second point . Let be the circle with diameter . Circles and meet at and a second point . Then , where and are relatively prime positive integers. Find .
Solution 1
Use the angle bisector theorem to find , , and use the Stewart's Theorem to find . Use Power of the Point to find , and so . Use law of cosines to find , hence as well, and is equilateral, so .
I'm sure there is a more elegant solution from here, but instead we do'll some hairy law of cosines:
(1)
Adding these two and simplifying we get:
(2). Ah, but (since lies on ), and we can find using the law of cosines:
, and plugging in we get .
Also, , and (since is on the circle with diameter ), so .
Plugging in all our values into equation (2), we get:
, or .
Finally, we plug this into equation (1), yielding:
. Thus,
or The answer is .
Solution 2
Let , , for convenience. We claim that is a symmedian. Indeed, let be the midpoint of segment . Since , it follows that and consequently . Therefore, . Now let . Since is a diameter, lies on the perpendicular bisector of ; hence , , are collinear. From , it immediately follows that quadrilateral is cyclic. Therefore, , implying that is a symmedian, as claimed.
The rest is standard; here's a quick way to finish. From above, quadrilateral is harmonic, so . In conjunction with , it follows that . (Notice that this holds for all triangles .) To finish, substitute , , to obtain as before.
-Solution by thecmd999
See Also
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