# Difference between revisions of "2006 USAMO Problems/Problem 6"

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== Problem == | == Problem == | ||

− | Let <math>ABCD</math> be a quadrilateral, and let <math>E</math> and <math>F</math> be points on sides <math>AD</math> and <math>BC</math> respectively, such that <math>\ | + | |

+ | Let <math> \displaystyle ABCD </math> be a quadrilateral, and let <math> \displaystyle E </math> and <math> \displaystyle F </math> be points on sides <math> \displaystyle AD </math> and <math> \displaystyle BC </math>, respectively, such that <math> \displaystyle AE/ED = BF/FC </math>. Ray <math> \displaystyle FE </math> meets rays <math> \displaystyle BA </math> and <math> \displaystyle CD </math> at <math> \displaystyle S </math> and <math> \displaystyle T </math> respectively. Prove that the circumcircles of triangles <math> \displaystyle SAE, SBF, TCF, </math> and <math> \displaystyle TDE </math> pass through a common point. | ||

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== Solution == | == Solution == | ||

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+ | {{solution}} | ||

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== See Also == | == See Also == | ||

− | *[[2006 USAMO Problems]] | + | |

+ | * [[2006 USAMO Problems]] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=490691#p490691 Discussion on AoPS/MathLinks] | ||

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+ | [[Category:Olympiad Geometry Problems]] |

## Revision as of 20:57, 1 September 2006

## Problem

Let be a quadrilateral, and let and be points on sides and , respectively, such that . Ray meets rays and at and respectively. Prove that the circumcircles of triangles and pass through a common point.

## Solution

*This problem needs a solution. If you have a solution for it, please help us out by adding it.*