2010 AIME II Problems/Problem 14
Problem
Triangle with right angle at , and . Point on $\overbar{AB}$ (Error compiling LaTeX. Unknown error_msg) is chosen such that and . The ratio can be represented in the form , where , , are positive integers and is not divisible by the square of any prime. Find .
Solution
Let be the circumcenter of and let the intersection of with the circumcircle be . It now follows that . Hence is isosceles and .
Denote the projection of onto . Now . By the pythagorean theorem, . Now note that . By the pythagorean theorem, . Hence it now follows that,
This gives that the answer is .
See also
2010 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 15 | |
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All AIME Problems and Solutions |