2009 OIM Problems/Problem 4
Problem
Let be a triangle with . Let be the incenter of and the other point of intersection of the exterior bisector of angle with the circumcircle of . The line intersects for the second time the circumcircle of at point . Show that the circumcircles of triangles and are tangent to and , respectively.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
Clearly is the midpoint of arc . Let , intersect the circumcircle of at , respectively. It is well known that is a parallelogram. Therefore, , which implies BI tangent to the circumcircle of . Similarly, is tangent to the circumcircle of .