1994 IMO Problems/Problem 2
Let be an isosceles triangle with . is the midpoint of and is the point on the line such that is perpendicular to . is an arbitrary point on different from and . lies on the line and lies on the line such that are distinct and collinear. Prove that is perpendicular to if and only if .
Let and be on and respectively such that . Then, by the first part of the problem, . Hence, is the midpoint of and , which means that is a parallelogram. Unless and , this is a contradiction since and meet at . Therefore, and , so , as desired.