Euler line

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Let $ABC$ be a triangle, points $H, N, G, O, L$ as $\triangle ABC$'s orthocenter, nine-point center, centroid, circumcenter, De Longchamps point respectively, then these points are collinear(regardless of $\triangle ABC$'s shape). And the line passes through points $H, N, G, O, L$ is called as Euler line, which is named after Leonhard Euler.

An interesting property of distances between these points on the Euler line:

  • $OG:GN:NH\equiv2:1:3$

Construct an orthic triangle$\triangle H_AH_BH_C$, then Euler lines of $\triangle AH_BH_C$,$\triangle BH_CH_A$,$\triangle CH_AH_B$ concur at $\triangle ABC$'s nine-point center.

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