2006 IMO Shortlist Problems/G2
(Ukraine) Let be a trapezoid with parallel sides . Points and lie on the line segments and , respectively, so that . Suppose that there are points and on the line segment satisfying
Prove that the points , , , and are concyclic.
Since and are collinear, the condition is equivalent to the condition that lines , , and are concurrent. Let be the point of concurrence.
Let be the circumcircles of , respectively. Since , the line is tangent to at . Similarly, is tangent to at . It follows there is a dilation centered at which takes to . Let denote the image of under . Evidently, are the respective images of under .
Now, since is tangent to at , it follows that
But is the image of under the dilation , so these two lines are parallel. Hence
Therefore are concyclic, as desired. ∎
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.