Euler straight line

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The straight line passing through the point $H$ of intersection of the altitudes of a triangle, the point $S$ of intersection of its medians (the centroid), and the centre $O$ of the circle circumscribed to it. If the Euler line passes through a vertex of the triangle, then the triangle is either isosceles or right-angled, or both right-angled and isosceles. The segments of the Euler line satisfy the relation


This line was first considered by L. Euler (1765).


See also: Triangle centre


[a1] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
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Euler straight line. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by P.S. Modenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article