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Try this
Drag any orange dot on a vertex of the triangle. The three dots representing the three centers will always lie on the green Euler line.
In the 18th century, the Swiss mathematician Leonhard Euler noticed that three of the many
centers of a triangle are always
collinear,
that is, they always lie on a straight line. This line has come to be named after him  the Euler line.
(His name is pronounced the German way  "oiler").
The three centers that have this surprising property are the triangle's
centroid
,
circumcenter
and
orthocenter.
In the figure above (press 'reset' first if necessary) the centroid is the black middle point on the line.
The circumcenter is the magenta point on the left, and the orthocenter is the red point on the right.
As you drag any of the triangle's
vertices around, you can see that these points remain
collinear,
all lying on the green Euler line.
The three centers involved each have their own page describing them, but here is a brief overview:
Centroid
The centroid is the point where the three
medians
converge. In the figure above click on "show details of Centroid".
The medians (here colored black) are the lines joining a vertex to the midpoint of the opposite side.
See Centroid of a Triangle for more.
Circumcenter
The circumcenter is the point where the
perpendicular bisectors
of the triangle's sides converge.
In the figure above click on "Show details of Circumcenter".
The three perpendicular bisectors (here colored magenta) are the lines that cross each side of
the triangle at right angles exactly at their midpoint.
See Circumcenter of a Triangle for more.
Orthocenter
The orthocenter is the point where the three
altitudes
of the triangle converge.
In the figure above click on "Show details of Orthocenter".
The three altitudes (here colored red) are the lines that pass through a vertex
and are perpendicular to the opposite side.
See Orthocenter of a Triangle for more.
Equilateral Triangles
Another interesting fact is that in an
equilateral triangle,
where all three sides have the same length, all three centers are in the same place.
In the figure above, adjust the vertices to try and get all three centers to come together.
You will see that the triangle is equilateral.
Related triangle topics
General
Perimeter / Area
Triangle types
Triangle centers
Congruence and Similarity
Solving triangles
Triangle quizzes and exercises
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