The nine-point circle and the Euler line | plane geometry | advanced level

Episode 102.

The nine-point circle and the Euler line | plane geometry | advanced level.

Branch of mathematics: plane geometry.
Difficulty level: advanced.

Theorem. The radius of the nine-point circle is twice smaller than the circumcircle.
Theorem. The circumcenter $O$, the centroid $G$, the orthocenter $H$, and the nine-point center $O_9$ of a triangle lie on a single line (which is called the Euler line of that triangle). Furthermore, the points lie on that line in this order: $O$, $G$, $O_9$, $H$, and the ratios of the segments are $OG:GO_9:O_9H=2:1:3$.

Mathematics. Geometry. Plane geometry.
#Mathematics #Geometry #PlaneGeometry

tags:
mathematics,geometry,plane geometry,classical geometry,Euclidean geometry,triangle,line,circle,circumcircle,altitude,perpendicular,nine-point circle,midpoint,median,orthocenter,centroid,Euler,Euler line,nine-point center,homothety

The same video on YouTube:
https://youtu.be/eiWVvVFVnaE

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https://t.me/mathematical_bunker/126