Difference between revisions of "2004 IMO Shortlist Problems/G8"
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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 21:03, 5 July 2021
Problem
A cyclic quadrilateral is given. The lines and intersect at , with between and ; the diagonals and intersect at . Let be the midpoint of the side , and let be a point on the circumcircle of such that . Prove that are collinear.