2015 AIME II Problems/Problem 11
The circumcircle of acute has center . The line passing through point perpendicular to intersects lines and and and , respectively. Also , , , and , where and are relatively prime positive integers. Find .
Call the and foot of the altitudes from to and , respectively. Let . Notice that because both are right triangles, and . By , . However, since is the circumcenter of triangle , is a perpendicular bisector by the definition of a circumcenter. Hence, . Since we know and , we have . Thus, . .
Notice that , so . From this we get that . So , plugging in the given values we get , so , and .
Let . Drawing perpendiculars, and . From there, . Thus, . Using , we get . Now let's find . After some calculations with ~ , . Therefore, . .
Let . Extend to touch the circumcircle at a point . Then, note that . But since is a diameter, , implying . It follows that is a cyclic quadrilateral.
Let . By Power of a Point, The answer is .
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