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/
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