Various Triangle Centers
Below are several figures for various points related to triangles.
These include
the centroid,
the circumcenter,
the orthocenter,
the incenter,
the excenters,
and the Euler line (which is a line, rather than a
point can you trust anything I say?).
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functioning. The bright red points (A, B, and C) can be moved around
with the mouse and the figure will adjust accordingly.

The centroid of a triangle is the point at which the three medians
meet. A median is the line between a vertex and the midpoint of the
opposite side.




The three perpendicular bisectors of the sides of a triangle meet
at the circumcenter. The circumcenter is also the center of
the circle passing through the three vertices, which circumscribes
the triangle. This circle is sometimes called the circumcircle.



The orthocenter is the point of intersection of the altitudes of the
triangle, that is, the perpendicular
lines between each vertex and the opposite side.
When the orthocenter is combined with the three vertices, any one of the
points is the orthocenter of the other three. This same property holds
for the set of four points consisting of the incenter
and the three excenters. Such points are said to
form an
orthocentric
system. Any three points from an orthocentric system all have the same
ninepoint circle associated with them.




The incenter of the triangle is the point at which the
three bisectors of the interior angles of the triangle meet. This is also
the center of the inscribed circle, also called the incircle
of the triangle.


An excenter is a point at which the line bisecting one interior
angle meets the bisectors of the two exterior angles on the opposite side.
This is the center of a circle, called an excircle which is tangent
to one side of the triangle and the extensions of the other two sides.
Naturally, every triangle has three excenters and three excircles.


