Euler Line

The Euler Line is a line that passes through the orthocenter, centroid, and circumcenter of a triangle.
The Euler Line

This geometric construction links several significant points of a triangle.

In every triangle, there are three key points: the centroid (G), the circumcenter (O), and the orthocenter (H).

No matter the shape or size of the triangle, except for equilateral triangles, these three points are always collinear.

The Euler Line is named after the Swiss mathematician Leonhard Euler, one of the greatest mathematicians of all time, renowned for his contributions to numerous fields of mathematics.

A Practical Example

Let's consider any triangle ABC.

The Euler Line

First, identify the centroid.

The centroid (G) is the point where the medians of the triangle intersect.

The medians and the centroid

A median is a line segment that connects a vertex (A, B, C) of the triangle to the midpoint (MAB, MBC, MAC) of the opposite side.

Note: The centroid divides each median in a 2:1 ratio, with the longer segment stretching from the vertex (A, B, C) to the centroid (G).

Next, locate the circumcenter.

The circumcenter (O) is the point where the perpendicular bisectors of the triangle's sides intersect, passing through their respective midpoints.

The circumcenter

Note: The circumcenter (O) is also the center of the circumcircle, the circle that passes through all three vertices of the triangle.

Finally, find the orthocenter.

The orthocenter (H) is the intersection point of the triangle's altitudes.

The orthocenter

An altitude of a triangle is a perpendicular segment drawn from a vertex to the opposite side or its extension.

These three points, G, O, and H, are aligned.

The line that passes through these three points is known as the Euler Line.

The Euler Line

Observations

Here are some observations about the Euler Line:

  • On the Euler Line, the centroid (G) is located between the circumcenter (O) and the orthocenter (H).
    The Euler Line
  • The distance between the centroid and the orthocenter (GH) is twice the distance between the centroid and the circumcenter (GO). $$ GH = 2GO $$ Therefore, the distance between the circumcenter and the orthocenter (OH) is three times the distance between the centroid and the circumcenter (GO). $$ OH = 3GO $$
  • The Case of the Equilateral Triangle
    The Euler Line does not exist in equilateral triangles, as the orthocenter, centroid, and circumcenter coincide (H=I=G=O). Therefore, in equilateral triangles, these points are not aligned, and there is a bundle of infinite lines passing through the same point instead of a single line.
    The equilateral triangle does not have the Euler Line
  • Other Notable Points of the Triangle
    The Euler Line also passes through other notable points of the triangle, such as the Longchamps point, the Schiffler point, the Exeter point, and the Gossard point.
  • The Incenter
    In general, the incenter does not lie on the Euler Line, except in isosceles triangles where the incenter is aligned with the orthocenter, centroid, and circumcenter.
    The incenter and the Euler Line
  • The Case of Isosceles Triangles
    In isosceles triangles, the Euler Line coincides with the axis of symmetry of the triangle.

And so on.

 

 
 

Please feel free to point out any errors or typos, or share suggestions to improve these notes. English isn't my first language, so if you notice any mistakes, let me know, and I'll be sure to fix them.

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