Mock AIME 3 Pre 2005 Problems/Problem 14
Problem
Circles and are centered on opposite sides of line , and are both tangent to at . passes through , intersecting again at . Let and be the intersections of and , and and respectively. and are extended past and intersect and at and respectively. If and , then the area of triangle can be expressed as , where and are positive integers such that and are coprime and is not divisible by the square of any prime. Determine .
Solution
Let and be the centers of and respectively.
Let point be the midpoint of . Thus, and
Let and be the radii of circles and respectively.
Let and be the areas of triangles and respectively.
Since and , then , and
This means that . In other words, those three triangles are similar.
Since is the circumcenter of ,
then
Let be the height of to side
Then, , thus
Since is the height of to side , then using similar triangles,
. Therefore, . Solving for we have:
By similar triangles,
Using Heron's formula,
, where we have:
, thus
~Tomas Diaz. orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
Mock AIME 3 Pre 2005 (Problems, Source) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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