Difference between revisions of "Mock Geometry AIME 2011 Problems/Problem 12"
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Revision as of 18:08, 15 January 2013
Problem
A triangle has the property that its sides form an arithmetic progression, and that the angle opposite the longest side is three times the angle opposite the shortest side. The ratio of the longest side to the shortest side can be expressed as , where are positive integers, is not divisible by the square of any prime, and and are relatively prime. Find .
Solution
Let the triangle be , such that , and , so that . Construct two points and on such that .
Since , . Therefore, , and . Now .
Let . From Angle Bisector Theorem on with angle bisector , , so .
Therefore, , and so . Now we have , and .
Now we use Angle Bisector Theorem on with angle bisector . We have , so .
Finally, notice that , so .
Now, , so from the Quadratic Formula, (we neglect the negative root), and so .
Therefore, , and the answer is .