and Related Topics

A circle that
passes through each of the vertices of a polygon is called the **circumcircle**
of the polygon.The center of that
circle is the **circumcenter**.Both
seem to be truncations of the prefix “circumscribing” attached to the words
circle or center.The word circumcircle
was suggested in 1883 by W. H. H. Hudson in *Nature Magazine,* according to Jeff Miller’s web site on
the first use of mathematical terms.

Because the
circumcenter must be the same distance from each of the vertices, and because
each side of the polygon is a chord of the circle, it is easy to understand
that the circumcircle must be at the intersection of the perpendicular
bisectors of the sides of the polygon.
Not all polygons have a single circle that passes through all the
vertices. Those that do are said to be **circumscribable** polygons. **All**
triangles are circumscribable.

The radius of the circumscribing triangle is given by the formula R= a*b*c/(4*area)where a, b, and c are the three side lengths, and the area represents the area of the triangle. If the area is written using Heron's formula, then you can express the radius only in terms of the three side lengths.

Several important theorems about the circumcenter of a triangle and its properties are given below. Many of the images link to an outstanding educational website maintained by Antonio Gutierrez with interactive and animated proofs that make geometry proofs come alive. It is one of the very finest sites on the net for the student of elementary geometry, IMHO. After the site opens, click on “Geometry Problems” in the table at the left of the screen.

**Steiner's theorem**, pictured at right, gives a relationship between the radii
of five important circles in any triangle, the cirucmcircle, the incircle, and
the three **excircles** (circles tangent to the three sides (or sides
extended) of a triangle on the outside of the triangle. Stated in another way, it says the sum of
the radii of the excircles equals the sum of the incircle radius added to four
times the radius of the circumcircle. You
can see an animated proof of the theorem at the web site of Antonio Gutierrez
by clicking on the graphic, which came from his page. The theorem is named for Jakob Steiner a Swiss geometrician of the 19th Century.

**Carnot’s Theorem **
states that in any triangle, the sum of
the distances from the circumcenter to the three sides of the triangle will
equal the sum of the radius of the incircle plus the radius of the circumcircle. The distance from the circumcircle to a side
is considered negative if the segment lies completely outside the triangle. This theorem is named for Lazere
Carnot, the father of Sadi
Carnot whose work features prominently in the study of thermodynamics.