[size=100][size=150]Euler, on the hunt for something else, [i]accidentally[/i] discovered this fascinating fact about triangle centers. [/size][/size]
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 [color=#c27ba0]circumcenter[/color], [color=#6aa84f]orthocenter[/color], and [color=#3d85c6]centroid[/color] all fell on the same line!
Fun Fact:
The line that passes through the three distinguished points - [color=#a64d79]circumcenter[/color], [color=#6aa84f]orthocenter[/color], and [color=#3c78d8]centroid[/color] - is called the [b][color=#bf9000]Euler Line[/color][/b]. [br][br][size=85][i]There's a few other points on this line as well, but we will get to that in a moment.[/i][/size]
You mean to say that this is true for all triangles?
[size=150][b][br][br]I KNOW RIGHT![/b][/size]
Well... not all triangles.
The[b] [color=#bf9000]Euler line [/color][/b]exists for all triangles... except [u]one[/u] kind. As you watch the animation below, think back to the triangles from the previous activities. Was there ever a special case?
When [b]DON'T[/b] 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...[br][br]Center of the Nine Point Circle[br]Schiffler's Point [br]Exeter Point [br]Longchamps Point [br]Gossard Prospector[br]Incenter [color=#666666](only if the triangle is isosceles)[/color]