Euler line

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Let $\triangle ABC$ be a triangle with orthocenter $H$, nine-point center $N$, centroid $G$, circumcenter $O$ and De Longchamps point $L$. Then these points are collinear and the line passes through points $H, N, G, O, L$ is called the Euler line of $\triangle ABC$. It is named after Leonhard Euler.

Certain fixed ratios hold among the distances between these points:

  • $OG:GN:NH = 2:1:3$

Given the orthic triangle$\triangle H_AH_BH_C$ of $\triangle ABC$, the Euler lines of $\triangle AH_BH_C$,$\triangle BH_CH_A$, and $\triangle CH_AH_B$ concur at $N$, the nine-point center of $\triangle ABC$.

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