2011 USAJMO Problems/Problem 5
Revision as of 23:04, 26 April 2015 by Kingsave3166 (talk | contribs) (AC and DE are parallel is given, no need to prove it)
Problem
Points , , , , lie on a circle and point lies outside the circle. The given points are such that (i) lines and are tangent to , (ii) , , are collinear, and (iii) . Prove that bisects .
Solution
Let be the center of the circle, and let be the midpoint of . Let denote the circle with diameter . Since , , , and all lie on .
Since quadrilateral is cyclic, . Triangles and are congruent, so , so . Because and are parallel, lies on (using Euler's Parallel Postulate). The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.