Euler Line
Meet Leonhard Euler (pronounced "oil-er").
Looking back at the triangles from the activities, what information did the triangles and their centers have to tell us?
Well, the circumcenter, orthocenter, and centroid all fell on the same line!
Fun Fact:
The line that passes through the three distinguished points - circumcenter, orthocenter, and centroid - is called the Euler Line.
There's a few other points on this line as well, but we will get to that in a moment.
You mean to say that this is true for all triangles?
Well... not all triangles.
The Euler line exists for all triangles... except one kind. As you watch the animation below, think back to the triangles from the previous activities. Was there ever a special case?
When DON'T these three points create a line?
But wait, there's more!
The three points we talked about aren't the only ones on the Euler line. Aside from the orthocenter, circumcenter, and centroid, some other points that fall on this line are...
Center of the Nine Point Circle
Schiffler's Point
Exeter Point
Longchamps Point
Gossard Prospector
Incenter (only if the triangle is isosceles)