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### Centroid, Circumcenter, Incenter, Orthocenter: Etymologies

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Date: 01/20/2002 at 20:20:21
From: meghan
Subject: Euler's line theorem

Why are the points centroid, circumcenter, orthocenter, and incenter
named as they are, and are there any other special points associated
with triangles?
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Date: 01/21/2002 at 10:10:26
From: Doctor Sarah
Subject: Re: Euler's line theorem

Hi Meghan - thanks for writing to Dr. Math.

From Steven Schwartzman's _The Words of Mathematics - An Etymological
Dictionary of Mathematical Terms Used in English_ (1994, Mathematical
Association of America):

center
from Greek kentron "a sharp point, a peg, a stationary point."
In times ancient and modern, people have put a stake in the ground and
attached an animal to the stake with a rope. The places that the
animal could wander all lie within a circle having the stake as its
center. The meaning of the word center was later abstracted away from
the stake as a pointed object, and the word came to mean the position
of the "stake" equidistant from all points on the circle.

centroid
from center and the Greek-derived -oid "looking like."
Metaphorically speaking a centroid looks like a center. Actually, it
doesn't so much look like a center as behave like one. An irregular
shape doesn't really have a center in the same sense that a symmetric
shape does; nor do real-world physical objects. Nevertheless, a
physical object has a point at which all the mass of the object seems
to be concentrated. Since that point acts like a center, it is called
a centroid.

circumcenter
from Latin circum "around," from the Indo-Eudopean root (s)ker- "to
bend," plus center.
With regard to a triangle, the circumcenter is the center of the
circumscribed circle. The circumcenter isn't necessarily "centered"
inside the triangle: the circumcenter of a right triangle is on the
hypotenuse, and the circumcenter of an obtuse triangle is outside the
triangle.

incenter
from in and center.
With regard to a triangle, the incenter is the center of the circle
(known as the incircle) that can be inscribed in that triangle.
Because the incenter is the point at which the bisectors of the three
angles of the triangle meet, the incenter is necessarily inside the
triangle.

orthocenter
the first component is from Greek orthos "straight, upright," hence
"perpendicular, from the Indo-European root wrodh- "to grow straight,
upright"; the second component is center.
With regard to a triangle, the place where the three altitudes (which
are perpendicular to the sides) meet is called the orthocenter... The
orthocenter is "centered" inside the triangle only when the triangle
is acute.

For more about triangle centers, see Clark Kimberling's Encyclopedia
of Triangle Centers, a searchable catalogue of hundreds of special
triangle points (centroid, incenter, Hofstadter points, Yff centers,
Apollonius point, etc.):

http://cedar.evansville.edu/~ck6/encyclopedia/

- Doctor Sarah, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Definitions
High School Geometry
High School Triangles and Other Polygons
Middle School Definitions
Middle School Geometry
Middle School Triangles and Other Polygons

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