2011 USAJMO Problems/Problem 5
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 .
Solutions
Solution 1
Let be the center of the circle, and let be the intersection of and . Let be and be .
, ,
Thus is a cyclic quadrilateral and and so is the midpoint of chord .
~pandadude
Solution 2
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).
Solution 3
Note that by Lemma 9.9 of EGMO, is a harmonic bundle. We project through onto , Where is the point at infinity for parallel lines and . Thus, we get , and is the midpoint of . ~novus677