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Difference between revisions of "Euler straight line"

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(See also Triangle centre)
 
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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, 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
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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 [[Median (of a triangle)|median]]s (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
  
 
$$OH:SH=1:2$$
 
$$OH:SH=1:2$$
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====Comments====
 
====Comments====
 
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See also: [[Triangle centre]]
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1963)</TD></TR></table>

Latest revision as of 20:16, 16 January 2016


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

$$OH:SH=1:2$$

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


Comments

See also: Triangle centre

References

[a1] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
How to Cite This Entry:
Euler straight line. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_straight_line&oldid=32621
This article was adapted from an original article by P.S. Modenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article