2021 USAMO Problems/Problem 6

Problem 6

Let $ABCDEF$ be a convex hexagon satisfying $\overline{AB} \parallel \overline{DE}$ , $\overline{BC} \parallel \overline{EF}$ , $\overline{CD} \parallel \overline{FA}$ , and $AB \cdot DE = BC \cdot EF = CD \cdot FA.$ Let $X$ , $Y$ , and $Z$ be the midpoints of $\overline{AD}$ , $\overline{BE}$ , and $\overline{CF}$ . Prove that the circumcenter of $\triangle ACE$ , the circumcenter of $\triangle BDF$ , and the orthocenter of $\triangle XYZ$ are collinear.

Solution

We construct two equal triangles, prove that triangle $XYZ$ is the medial triangle of both this triangles, use property of medial triangle and prove that circumcenters of constructed triangles coincide with given circumcenters.

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2021 USAMO Problems/Problem 6

Problem 6

Solution