Talk:2021 USAMO Problems/Problem 1
We are given the acute triangle , rectangles such that . Let's call .
Construct circumcircles around the rectangles respectively. intersect at two points: and a second point we will label . Now is a diameter of , and is a diameter of , so , and , so is on the diagonal .
(angles standing on the same arc of the circle ), and similarly, . Therefore, .
Construct another circumcircle around the triangle , which intersects in , and in . We will prove that . Note that is a cyclic quadrilateral in , so since , is a diameter of and - so is a rectangle.
Since , is a diameter of , and . Similarly, , so O is on , and by opposite angles, .
Finally, since is on and is on , that gives - meaning is on the line . But is also on the line - so . Similarly, . So the three diagonals intersect in .