2000 IMO Problems/Problem 1
Two circles and intersect at two points and . Let be the line tangent to these circles at and , respectively, so that lies closer to than . Let be the line parallel to and passing through the point , with on and on . Lines and meet at ; lines and meet at ; lines and meet at . Show that .
Given a triangle, and a point in it's interior, assume that the circumcircles of and are tangent to . Prove that ray bisects . Let the intersection of and be . By power of a point, and , so .
Let ray intersect at . By our lemma, , bisects . Since and are similar, and and are similar implies bisects .
By simple parallel line rules, . Similarly, , so by criterion, and are congruent.
Now, and since is parallel to . But is tangent to the circumcircle of hence and that implies So is isosceles and .
Join points and , is perpendicular on (why?), previously we proved , hence is isoscles and .