Mock AIME 3 Pre 2005 Problems/Problem 11
is an acute triangle with perimeter . is a point on . The circumcircles of triangles and intersect and at and respectively such that and . If , then the value of can be expressed as , where and are relatively prime positive integers. Compute .
Remark that since is cyclic we have , and similarly . Therefore by AA similarity . Thus there exists a spiral similarity sending to and to , so by a fundamental theorem of spiral similarity . The angle equality condition gives , so is isosceles and . Similarly, . Finally, note that the congruent side lengths actually imply , so .
Let and . Remark that from the perimeter condition . Now from Power of a Point we have the system of two equations Expanding the second equation and rearranging variables gives . Back-substitution yields and consequently . Thus and , so the desired ratio is .
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