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Difference between revisions of "2004 IMO Shortlist Problems/G8"

(Created page with "A cyclic quadrilateral <math>ABCD</math> is given. The lines <math>AD</math> and <math>BC</math> intersect at <math>E</math>, with <math>C</math> between <math>B</math> and <m...")
 
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== Problem ==
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A cyclic quadrilateral <math>ABCD</math> is given. The lines <math>AD</math> and <math>BC</math> intersect at <math>E</math>, with <math>C</math> between <math>B</math> and <math>E</math>; the diagonals <math>AC</math> and <math>BD</math> intersect at <math>F</math>. Let <math>M</math> be the midpoint of the side <math>CD</math>, and let <math>N \neq M</math> be a point on the circumcircle of <math>\triangle ABM</math> such that <math>\frac{AN}{BN} = \frac{AM}{BM}</math>. Prove that <math>E, F, N</math> are collinear.
 
A cyclic quadrilateral <math>ABCD</math> is given. The lines <math>AD</math> and <math>BC</math> intersect at <math>E</math>, with <math>C</math> between <math>B</math> and <math>E</math>; the diagonals <math>AC</math> and <math>BD</math> intersect at <math>F</math>. Let <math>M</math> be the midpoint of the side <math>CD</math>, and let <math>N \neq M</math> be a point on the circumcircle of <math>\triangle ABM</math> such that <math>\frac{AN}{BN} = \frac{AM}{BM}</math>. Prove that <math>E, F, N</math> are collinear.
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[[Category:Olympiad Geometry Problems]]

Revision as of 22:03, 5 July 2021

Problem

A cyclic quadrilateral $ABCD$ is given. The lines $AD$ and $BC$ intersect at $E$, with $C$ between $B$ and $E$; the diagonals $AC$ and $BD$ intersect at $F$. Let $M$ be the midpoint of the side $CD$, and let $N \neq M$ be a point on the circumcircle of $\triangle ABM$ such that $\frac{AN}{BN} = \frac{AM}{BM}$. Prove that $E, F, N$ are collinear.

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