2021 USAMO Problems/Problem 6
Let be a convex hexagon satisfying , , , andLet , , and be the midpoints of , , and . Prove that the circumcenter of , the circumcenter of , and the orthocenter of are collinear.
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.
Similarly 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
Similarly 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 circumcenter coincide with circumcenter
Similarly circumcenter of coincide with circumcenter of
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