The Euler line of a triangle

The Euler Line of a triangle is simply a straight line that passes through four of the commonly known centers of the triangle. They are, 

Circumcenter(C)

Centroid(G)

Orthocenter(H)

Nine-point center(L)

It creates more of an impact if we say it the other way around. That is, for any given triangle the above mentioned four centers are always collinear; they lie on the same line - the Euler Line. 

Things are slightly different for an equilateral triangle. These four centers are the same single point. We can draw infinitely many lines passing through a single point. And so it only makes sense that the concept of 'the' Euler line is associated only with non-equilateral triangles. 

Now for those of you reading this who don't know about these four centers of a triangle, let me quickly give a short description of each one of them. 

Circumcenter is the center of the circumcircle of a triangle. It is the circle through the three vertices of the triangle. Circumcenter also happens to be the point where all three perpendicular bisectors of the edges of the triangle intersect. 

More on Circumcenter here.

Centroid is the point where all three medians of a triangle intersect. A median of a triangle is a segment connecting a vertex to the midpoint of the opposite side. Every triangle has three medians which always intersect at a single point called centroid.

More about Centroid here.

Orthocenter is the point where all three altitudes of a triangle intersect. 

More here.

Nine-point center is the center of the nine-point circle of a triangle. It is the circle that passes through nine significant points of the triangle. These nine points are the feet of the three altitudes, midpoints of the three sides(or the points where medians meet the three sides), midpoints of the line segments from the vertices of the triangle to the orthocenter. 

It is not only that circumcenter, orthocenter, centroid and nine-point center lie on the same line, their distances from one another are also related. For example, in a triangle the centroid is twice as far from the orthocenter than it is from the circumcenter. GH = 2GC. 

The nine-point center is exactly in the middle of the circumcenter and the orthocenter. LH = LC.

Using these two relations we can relate the other distances. For example, 

If GC = x

   GH = 2GC = 2x

   CH = 3x = 3GC

   CH = \(\frac{3}{2}\)GH

So from the last two steps, the distance between the circumcenter and orthocenter is thrice the distance between the centroid and circumcenter, and three halves of the distance between the centroid and orthocenter. 

Also, LG = LC = 2CH = \(2 \cdot \frac{3}{2}\)GH

So, the distance between the nine-point center and centroid is thrice the distance between the centroid and orthocenter. The distance between the nine-point center and circumcenter is also thrice the distance between the centroid and orthocenter.  

Read : Constructing the Euler Line in Geogebra