A triangle has no one unique center, but the circumcenter may be the second most popular and easy to visualize, after the incenter.

The circumcircle is the smallest circle that can fit through the three points that define a triangle. The circumcircle has a radius, R, that is equal to a*b*c/(4K), where K is the area of the triangle, and a, b, and c are the side lengths of the triangle ΔABC. We will denote the circumcenter as O

.

The circumcenter's coordinates are:

```       (dA*(Cy-By) + dB*(Ay-Cy) + dC*(By-Ay))
Ox = -----------------------------------------   (1a)
2*(Ax*(Cy-By) + Bx*(Ay-Cy) + Cx*(By-Ay))

-(dA*(Cx-Bx) + dB*(Ax-Cx) + dC*(Bx-Ax))
Oy = -----------------------------------------   (1b)
2*(Ax*(Cy-By) + Bx*(Ay-Cy) + Cx*(By-Ay))
```

where:

```O = (Ox, Oy)... Cartesian coordinates of the circumcenter
A = (Ax, Ay)... the coordinates of vertex A of triangle ABC
B = (Bx, By)... the coordinates of vertex B of triangle ABC
C = (Cx, Cy)... the coordinates of vertex C of triangle ABC
```

and where some intermediate calculations help reduce eyestrain:

```dA  = Ax^2 + Ay^2           (2a)
dB  = Bx^2 + By^2           (2b)
dC  = Cx^2 + Cy^2           (2c)
```

The best explanation for finding the center of the circle is found here. The Khan Academy always has marvelous tutorials on YouTube, and they also explain this quite well here..

Example 1: An acute triangle has vertices A, B, and C at A = (-2,-2), B = (5,3), and C = (1,4). Find the circumcenter O and the radius of the circumcircle, R.
Solution:

``` A  = (Ax, Ay) = (-2,-2)
B  = (Bx, By) = ( 5, 3)
C  = (Cx, Cy) = ( 1, 4)
dA  = (-2)^2 + (-2)^2 =  8
dB  = 5^2 + 3^2       = 34
dC  = 1^2 + 4^2       = 17
O = (Ox, Oy) = (2.06, -0.28)
R  = 4.4
```

The result is that the circumcenter is found at (2.06, -0.28) and the radius of the circumcircle is 4.4.

Point of Concurrency of Perpendicular Bisectors: The circumcenter is the point of concurrency of the perpendicular bisectors of each side. If you bisect every side, and you draw the line that runs perpendicular to that side then every line intersects at one point: the circumcenter.

When I began this writeup, I was under the impression that Euclid had proved that the perpendicular bisectors from every side all meet at the same point for every possible triangle. He showed something similar in Book 4, Proposition 5 of The Elements. But Cut the Knot mathematician and author Alexander Bogomolny says that Euclid didn't do this. He only showed that they did, but offered no proof. It's Bogomolny's belief that Euclid would have needed a Ceva's Theorem in order to prove it, but that theorem didn't come along for another 1500 years.

SOURCE: Jim Wilson, Proof that the three perpendicular biectors of the sides of a triangle are concurrent. Wilson is a professor with the Mathematics Education program at the University of Georgia. His web site is full of mathematical topics.

The Circumcenter is Outside the Triangle for Obtuse Triangles: Although the incenter is always inside the triangle, the circumcenter does not have to be. When the triangle is acute, the circumcenter is inside ΔABC. When it is obtuse, O is outside. Example 2 gives points of a very small obtuse angle with a wide vertex angle at A. The circumcenter is a large distance away from the triangle.

Example 2: An obtuse triangle has one vertex at the origin. We'll label this vertex A. The triangle is isosceles, meaning that sides AB and AC are of equal length. The interior angle α is 150°. The vertex coordinates are: A = (0,0), B = (1,0), and C = (cos(150°),sin(150°)) = (-0.87, 0.5). Find the circumcenter and the radius of the circumcircle.
Solution:

``` A  = (Ax, Ay) = ( 0,    0)
B  = (Bx, By) = ( 1,    0)
C  = (Cx, Cy) = (-0.87, 0.5)
dA  =   3.73
dB  =   1
dC  =   1
O = (Ox, Oy) = (0.5, 1.87)
R  = 1.93
```

The circumcenter is located at O = (0.5, 1.87). The radius is R = 1.93, a comparatively large distance away from the triangle. This is for an interior angle α = 150°. If the interior angle were greater, the radius would be even larger. For α = 170°, R = 5.7.

A straightforward equation for the circumcenter was difficult to find on the internet, and when I sat down to derive it myself, I was dismayed at how messy the terms got. When I did find a workable equation (Equation 1 above), I wanted to see if it would work for a variety of cases, and so I dropped the equation and many of its preceding calculational terms into Excel, created a graph of a triangle, the circumcenter point, and then the circumcircle to see if the equation would work and was well behaved and so forth. A picture of Example 1 is on my homenode, and will stay there for a brief time.

Barycentric Coordinates: The barycentric coordinates of the circumcenter are sin(2α):sin(2β):sin(2γ). (The interior angles at triangle vertices A, B, and C are α, β, γ, respectively.)

Trilinear Coordinates: The trilinear coordinates of the circumcenter are cos(α):cos(β):cos(γ).

Everything2 Writeups: Articles on (topic)

1. tongpoo, circumcenter, Dec. 2, 2001
2. tongpoo, circumcircle. A nodeshell was created, but it was never filled. Clearly this hole must be filled!
3. tongpoo, triangle, Feb. 8, 2002
4. IWhoSawTheFace, incenter, Feb. 8, 2002

References: Useful books and references on geometry

1. H.S.M. Coxeter, Introduction to Geometry, 2nd Ed., (c) 1969
*SIGH* What a magnificent book.
§ 1.4, “The Medians and the Centroid,” p. 10
§ 1.5, “The Incircle and the Circumcircle,” pp. 11-16
§ 1.6, “The Euler Line and the Orthocenter,” p. 17
2. Dan Pedoe, Geometry: A Comprehensive Course
3. J.L. Heilbron, Geometry Civilized, ©2000
4. David Wells, Ed., The Penguin Dictionary of Curious and Interesting Geometry, ©1991
5. Bruce Meserve, Fundamental Concepts of Geometry, ©1983
6. Melvin Hausner, A Vector Space Approach to Geometry, ©1965
7. Gerald Farin and Dianne Hansford, The Geometry Toolbox, ©1998
Ch. 3, 2D Lines
§ 3.6, “Distance of a point to a line,” p. 40
§ 3.7, “The foot of a point,” p. 44
§ 3.8, “Computing intersections,” p. 45
Ch. 8, Breaking it up: Triangles
§ 8.1, “Barycentric coordinates,” p. 126
§ 8.2, “Affine invariance,” p. 128
§ 8.3, “Some special points,” p. 128
8. Daniel Zwillinger, Ed., The CRC Standard Mathematical Tables and Formulae, 30th Ed, ©1996
Ch. 4, Geometry,
§ 4.5.1, “Triangles,” p. 271
§ 4.6, “Circles,” p. 277
9. Siobhan Roberts, King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry, ©2006
10. Alfred S. Posamentier and Charles T. Salkind, Challenging Problems in Geometry, ©1988
11. Hans Rademacher and Otto Toeplitz, The Enjoyment of Mathematics, Published in 1957 by the Princeton University Press
§ 26, “A characteristic property of the circle,” p. 160
§ 28, “The indispensability of the compass for the constructions of elementary geometry,” p. 177

Internet References

1. Wikipedia, "Circumscribed Circle" This contains very useful mathematical formulae, especially matrix forms for finding the center of the circle, and exterior angles at the intersections of the circumcircle with the vertices of a triangle.
2. Wikipedia, "Triangle"
3. Wikipedia, "Incircle and Excircles of a Triangle"
4. D. Joyce, Euclid's Elements, Book 4, Proposition 5, "To circumscribe a circle about a given triangle." David Joyce is a professor of Mathematics and Computer Science at Clark University. He rendered Euclid's Elements into HTML, added Java applets to illustrate geometric constructions with live, movable points and lines. If you're a geometry buff, you should bookmark this site.
5. Jim Wilson, Proof that the three perpendicular biectors of the sides of a triangle are concurrent. Wilson is a professor with the Mathematics Education program at the University of Georgia. His web site is full of mathematical topics.
6. To construct a circle given three points. Nice Java applet allows you to drag vertices around and watch the circumcenter move.
7. Weisstein, Eric W. "Circumcenter" From MathWorld--A Wolfram Web Resource.
8. Weisstein, Eric W. "Circumcircle" From MathWorld--A Wolfram Web Resource.
9. Weisstein, Eric W. "Circumradius" From MathWorld--A Wolfram Web Resource.
10. Weisstein, Eric W. "Incenter" From MathWorld--A Wolfram Web Resource.
11. Weisstein, Eric W. "Triangle" From MathWorld--A Wolfram Web Resource.
12. Weisstein, Eric W. "Tangential Triangle" From MathWorld--A Wolfram Web Resource.
13. Alexander Bogomolny, "Incircle and Excircles of a Triangle" From Cut The Knot--mathematical topics. Cut the Knot has a full range of geometric topics.