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Difference between revisions of "2021 USAMO Problems/Problem 6"

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==Problem 6 ==
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Let <math>ABCDEF</math> be a convex hexagon satisfying <math>\overline{AB} \parallel \overline{DE}</math>, <math>\overline{BC} \parallel \overline{EF}</math>, <math>\overline{CD} \parallel \overline{FA}</math>, and<cmath>AB \cdot DE = BC \cdot EF = CD \cdot FA.</cmath>Let <math>X</math>, <math>Y</math>, and <math>Z</math> be the midpoints of <math>\overline{AD}</math>, <math>\overline{BE}</math>, and <math>\overline{CF}</math>. Prove that the circumcenter of <math>\triangle ACE</math>, the circumcenter of <math>\triangle BDF</math>, and the orthocenter of <math>\triangle XYZ</math> are collinear.
  
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==Solution==
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We construct two equal triangles, prove that triangle <math>XYZ</math> is the medial triangle of both this triangles, use property of medial triangle and prove that circumcenters of constructed triangles coincide with given circumcenters.

Revision as of 06:33, 15 September 2022

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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