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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Euler line