Difference between revisions of "Mock Geometry AIME 2011 Problems/Problem 5"
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Revision as of 21:52, 1 January 2012
Problem
In triangle The bisector of angle meet at and the circumcircle at different from . Calculate the value of
Solution
because they are both subscribed by arc . because they are both subscribed by arc . Hence , because . Then is isosceles.
Let be the foot of the perpendicular from to . As is isosceles, it follows that is the midpoint of , and so . From the angle bisector theorem, . We have . Solving this system of equations yields . Thus, .
because they are vertical angles. It was shown , and so by similarity. Then and so .
Then by the Pythagorean Theorem on , . Also from , . Subtracting these equations yields , and so .