2004 IMO Shortlist Problems/G8
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.
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,