Circumcircle, Circumcenter
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.