# 2021 USAMO Problems/Problem 6

## 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.

## 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

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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