Question 6
Let $ABCDEF$ be a convex hexagon with $AB\parallel DE$, $BC\parallel EF$, $CD\parallel FA$, and
$$AB\cdot DE=BC\cdot EF=CD\cdot FA.$$
Let $X,Y,Z$ be the midpoints of $AD,BE,CF$. Prove that the circumcenters of $ACE$ and $BDF$ and the orthocenter of $XYZ$ are collinear. The original statement is retained unchanged in scope; see the explicit degeneracy caveat in the solutions.
Proposed by Ankan Bhattacharya. Evan Chen’s third-party solution notes: https://aops.com/community/p21498548; source video https://youtu.be/9WNgDETHOlI?t=2930. One point is nonofficial editorial normalization; manual essay grading only.
$$AB\cdot DE=BC\cdot EF=CD\cdot FA.$$
Let $X,Y,Z$ be the midpoints of $AD,BE,CF$. Prove that the circumcenters of $ACE$ and $BDF$ and the orthocenter of $XYZ$ are collinear. The original statement is retained unchanged in scope; see the explicit degeneracy caveat in the solutions.
Proposed by Ankan Bhattacharya. Evan Chen’s third-party solution notes: https://aops.com/community/p21498548; source video https://youtu.be/9WNgDETHOlI?t=2930. One point is nonofficial editorial normalization; manual essay grading only.