2023 USAJMO Problems/Problem 2
Problem
(Holden Mui) 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
.
Solution 1
The condition is solved only if is isosceles, which in turn only happens if
is perpendicular to
.
Now, draw the altitude from to
, and call that point
. Because of the Midline Theorem, the only way that this condition is met is if
, or if
.
By similarity,
. Using similarity ratios, we get that
. Rearranging, we get that
. This implies that
is cyclic.
Now we start using Power of a Point. We get that , and
from before. This leads us to get that
.
Now we assign variables to the values of the segments. Let and
. The equation from above gets us that
. As
from the problem statements, this gets us that
and
, and we are done.
-dragoon and rhydon516 (: