2023 USAMO Problems/Problem 1
In an acute triangle , let be the midpoint of . Let be the foot of the perpendicular from to . Suppose that the circumcircle of triangle intersects line at two distinct points and . Let be the midpoint of . Prove that .
Let be the foot from to . By definition, . Thus, , and .
From this, we have , as . Thus, is also the midpoint of .
Now, iff lies on the perpendicular bisector of . As lies on the perpendicular bisector of , which is also the perpendicular bisector of (as is also the midpoint of ), we are done. ~ Martin2001
We are going to use barycentric coordinates on . Let , , , and , , . We have and so and . Since , it follows that Solving this gives so The equation for is Plugging in and gives . Plugging in gives so Now let where so . It follows that . It suffices to prove that . Setting , we get . Furthermore we have so it suffices to prove that which is valid.
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