USAMO 2026 (Problem 5)
Let be a triangle. Points , , and lie on sides , , and , respectively, such that
Let , , and be the circumcenters of triangles , , and , respectively. Let , , and be the circumcenters of triangles , , and , respectively. Prove that .
By Miquel's theorem, the circumcircles of , , and are concurrent at .
Prove that
with spiral similarity centered at .
By (1) and (2),
Therefore,
So,
with spiral similarity centered at .
Similarly,
with all spiral similarities centered at .
Prove that , with spiral similarity centered at .
So,
and are the circumcenters of and , respectively.So, under the same direct spiral similarity centered at (),
Therefore,
Similarly,
Prove that , with spiral similarity centered at .
Therefore,
Hence,
with spiral similarity centered at .
Prove that , with spiral similarity centered at .
, , and are the circumcenters of , , and , respectively. Therefore,
Let .
In , , so
Similarly,
Therefore,
By (1) and (2),
with spiral similarity centered at .
Let's consider the complex plane and, without loss of generality, assume that is the origin.By Spiral Similarity and Hint 6,
There is such that
By Spiral Similarity and Hint 7,
There is such that
To prove , prove .