Difference between revisions of "2004 IMO Shortlist Problems/G8"
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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> | 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. | ||
Revision as of 00:41, 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.
is on by Lemma 9.17 on Euclidean Geometry in Maths Olympiad. 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,