Euler line

Euler line(Euler plug), triangle straight line, as for the name the mathematician who discovers existence レオンハルト Euler.

Table of contents


In the triangle in the figure above

  • Intersection of line of blue orthocenter
  • Intersection of yellow line center of gravity
  • Intersection of line of green circumcenter

The red line where it passes by these points is the Euler line.

Circumcenter O Center of gravity G Orthocenter H Between always, 2OG = GH Relationship has consisted.

Proof of existence in straight line

The method of proving the fact that these three points are on the identical straight line is listed some sort.

Analytical method
  • Triangle coordinate plane surface, seek the coordinate of three points and you show the fact that it is on the identical straight line.
Geometric method
  • The line which ties circumcenter and the orthocenterMedium lineThe fact that the intersection is center of gravity is shown.
  • When triangle expanding to two times center of gravity as a center, the fact that the tip of moving the circumcenter is the orthocenter of original triangle is shown.
The method of using vector
  • AH = OB + OC, G = (A + B + C) /3 and the like is utilized.
The method of using trilinear coordinateses barycentric coordinates
  • To display the circumcenter center of gravity orthocenter in the above-mentioned coordinate, that queue system becomes 0 is shown.

Special point on line

Several important point other than the circumcenter center of gravity orthocenter which is on the Euler line is listed.

nine point circle
In triangle
  • Three sidesPoint
  • The foot of the perpendicular line which was lowered to the opposite side from three apexes
  • Point of orthocenter and apex
It passes by these nine points circleNine point circlesWith it calls. The center of this circle hits to the point of circumcenter and the orthocenter.
ド ロンシャン point
Vis-a-vis circumcenter the orthocenter the point which is symmetrical positionド ロンシャン pointWith you say. This point L Distance,
AL2 - BC2 = BL2 - CA2 = CL2 - AB2

Euler line of special triangle

Right triangle
Right triangleThe Euler line becomes the line which ties the point of apex and the hypotenuse which are perpendicular. As for this you understand easily from the fact that thing and the orthocenter where circumcenter is point in hypotenuse are apex.
Isosceles triangles
Isosceles trianglesThe Euler line becomes the medium line in angle. Because as for this because this straight line has entirely the character below, the circumcenter center of gravity orthocenter comes on this straight line.
In addition from 4th character, it understands that also the incenter is on the identical line.
Regular triangle
Because the circumcenter center of gravity orthocenter agrees, it cannot define the Euler line.
Excenter triangle
The triangle which three excenters of triangle make is called excenter triangle. The Euler line of this triangle becomes circumcenter of original triangle and the straight line which ties the incenter.


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