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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+ | == Solution == | ||
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+ | Let <math>P = CD \cap EF</math>. Let <math>Q = AB \cap CD</math>. Let <math>R = AB \cap EF</math>. Let<math>(ABM)</math> denote the circumcircle of <math>\triangle ABM</math>. Let <math>N' = EF \cap (ABM)</math>. Note that <math>N' = PR \cap (ABM)</math> | ||
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+ | Claim: <math>P</math> is on <math>(ABM)</math>. Proof: <math>(C, D; P, Q) = -1</math> as complete quadrilaterals induce harmonic bundles. <math>QP \cdot QM = QC \cdot QD</math> by Lemma 9.17 on Euclidean Geometry in Maths Olympiad. By power of a point theorem, <math>QP \cdot QM = QA \cdot QB</math> and this is equivalent to our original claim. | ||
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+ | <math>(A, B; R, Q) = -1</math> as complete quadrilaterals induce harmonic bundles. By a projection through <math>P</math> from <math>AQ</math> onto <math>(ABM)</math>, <math>(A, B; N', M) = -1</math>. Since <math>\frac{AN}{BN} = \frac{AM}{BM}</math>, <math>M</math> and <math>N</math> are on the intersections of <math>(ABM)</math> and an Appollonian circle centered on AB, so N and M are on the opposite sides of AB. Therefore, <math>(A, B; N, M) = -\frac{AN}{BN}\div\frac{AM}{BM} = -1</math>. By uniqueness of harmonic conjugate, <math>N = N'</math> | ||
[[Category:Olympiad Geometry Problems]] | [[Category:Olympiad Geometry Problems]] |
Latest revision as of 00:43, 6 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.
Solution
Let . Let . Let . Let denote the circumcircle of . Let . Note that
Claim: is on . Proof: as complete quadrilaterals induce harmonic bundles. by Lemma 9.17 on Euclidean Geometry in Maths Olympiad. By power of a point theorem, and this is equivalent to our original claim.
as complete quadrilaterals induce harmonic bundles. By a projection through from onto , . Since , and are on the intersections of and an Appollonian circle centered on AB, so N and M are on the opposite sides of AB. Therefore, . By uniqueness of harmonic conjugate,