Centroid, Circumcenter, Incenter, Orthocenter: Etymologies

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?

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/