2001 IMO Shortlist Problems/G1
Let be the center of the square inscribed in acute triangle with two vertices of the square on side . Thus one of the two remaining vertices of the square is on side and the other is on . Points are defined in a similar way for inscribed squares with two vertices on sides and , respectively. Prove that lines are concurrent.
Let , , , , , and . Let be the point on the other side of than such that is an isosceles right triangle. Define and similarly. Let and be the points on and that are the vertices of the square centered at . We then have that is also an isosceles right triangle. It's clear that is parallel to , so and . The ratio of similarity of both relations is , which implies that quadrilaterals and are similar. Therefore and . It then follows that , , and are collinear. Similarly , , and are collinear, as are , , and . It therefore suffices to show that , , and are concurrent.
Let and , for positive reals and . Also let and . Note that , and likewise . It then follows from the Law of Sines on triangles and that
Solving for and gives that
Similar lines of reasoning show that
An application of the trigonometric version of Ceva's Theorem shows that , , and are concurrent, which shows that , , and are concurrent.