2010 USAMO Problems/Problem 4
Let be a triangle with . Points and lie on sides and , respectively, such that and . Segments and meet at . Determine whether or not it is possible for segments to all have integer lengths.
We know that angle , as the other two angles in triangle add to . Assume that only , and are integers. Using the Law of Cosines on triangle BIC,
. Observing that and that , we have
Since the right side of the equation is a rational number, the left side (i.e. ) must also be rational. Obviously since is irrational, this claim is false and we have a contradiction. Therefore, it is impossible for , and to all be integers, which invalidates the original claim that all six lengths are integers, and we are done.