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>\frac{AE}{ED}=\frac{BF}{FC}.</math> Ray <math>FE</math> meets rays <math>BA</math> and <math>CD</math> at <math>S</math> and <math>T</math> respectively. Prove that the circumcircles of triangles  <math>SAE, SBF, TCF,</math> and <math>TDE</math> pass through a common point.
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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]]
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* [[2006 USAMO Problems]]
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* [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 19:57, 1 September 2006

Problem

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

Solution

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See Also

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