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 o 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
(given), and
(angles on the same arc). But since
is on
, that gives
- meaning
is on the line
. But
is also on the line
- so
. Similarly,
.
Finally, since ,
is a diameter of
, and
. Similarly,
, so O is on
, and the three diagonals are concurrent.