2009 AIME I Problems/Problem 15
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[hide]Problem
In triangle , , , and . Let be a point in the interior of . Let and denote the incenters of triangles and , respectively. The circumcircles of triangles and meet at distinct points and . The maximum possible area of can be expressed in the form , where , , and are positive integers and is not divisible by the square of any prime. Find .
Solution 1
First, by Law of Cosines, we have so .
Let and be the circumcenters of triangles and , respectively. We first compute Because and are half of and , respectively, the above expression can be simplified to Similarly, . As a result
Therefore is constant (). Also, is or when is or . Let point be on the same side of as with ; is on the circle with as the center and as the radius, which is . The shortest distance from to is .
When the area of is the maximum, the distance from to has to be the greatest. In this case, it's . The maximum area of is and the requested answer is .
Solution 2
From Law of Cosines on , Now, Since and are cyclic quadrilaterals, it follows that Next, applying Law of Cosines on , By AM-GM, , so Finally, and the maximum area would be so the answer is .
Solution 3
First, we notice that triangle ABC is a scaled version of a 5-7-8 triangle (which has a 60 degree angle opposite the side with length 7). So . Therefore, let and Therefore, in triangle , we know that and . Now note that quadrilaterals and are both cyclic. This means that and . Therefore, .
Now note that in order to maximize the area of , we have to maximize the distance from to line . Note that since the second intersection of the Circles must lie below line , we try to find the locus of all points P under BC such that . Let the circumcenter of triangle be . Then major $\arc{BC} = 2 * \angle{BCP} = 300^\circ{}$ (Error compiling LaTeX. Unknown error_msg), minor $\arc{BC} = 60^\circ{}$ (Error compiling LaTeX. Unknown error_msg) which means is an equilateral triangle. (I will work on finishing solution maybe sometime later today...)
See also
2009 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Question | |
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