Talk:2021 USAMO Problems/Problem 1
I spent some time poking around at this with the help of Geogebra, and after a few failed attempts (trying to figure out something from the relationship when
and using applications of the Cosine rule to try to get something in terms of the triangle's angles) I realized that the point of concurrency is the intersection of the circumcircles of the three rectangles. The main difficulty after that is proving that this point is also the intersection point of the diagonals.
Clearly, if we construct two rectangles on and
and then circumscribe them, the circles meet in two points,
and
. Equally clearly, the angles
and
.
Now we can construct a circumcircle for the triangle which will intersect the perpendiculars from
at
in
, and
. Now *if* the triples
are collinear, then the angle
and by construction
and we're done. But I can't figure out how to prove the collinearity (that is, to prove that the intersection of the circumcircles is on the diagonal).