2021 USAMO Problems/Problem 6
Problem 6
Let be a convex hexagon satisfying
,
,
, and
Let
,
, and
be the midpoints of
,
, and
. Prove that the circumcenter of
, the circumcenter of
, and the orthocenter of
are collinear.
Solution
We construct two equal triangles, prove that triangle is the same as medial triangle of both this triangles. We use property of medial triangle and prove that circumcenters of constructed triangles coincide with given circumcenters.
Denote
Then
Denote
Symilarly we get
The translation vector maps into
is
so is midpoint of
and
Symilarly
is the midpoint of
and
is the midpoint of
and
is the midpoint of
Symilarly is the midpoint of
is the midpoint of
Therefore is the medial triangle of
is
translated on
It is known (see diagram) that circumcenter of triangle coincide with orthocenter of the medial triangle. Therefore orthocenter of
is circumcenter of
translated on
It is the midpoint of segment connected circumcenters of
and
According to the definition of points quadrangles
and
are parallelograms. Hence
Power of points A,C, and E with respect circumcircle
is equal, hence distances between these points and circumcenter of
are the same. Therefore circumcenters of constructed triangles coincide with given circumcenters.
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