A triangle has exactly one unique circle that can fit through all of its vertices. This circle is termed the circumcircle by geometers.

Let the triangle's vertices be the points A, B, and C. Then the circumcircle is defined by these three points uniquely. The three points uniquely identify both the center and the radius of the circumcircle. The three points of intersection between the circumcircle and the triangle are the three vertices, A, B, and C.

A summary of key facts about the circumcircle:

• The circumcenter is the intersection of three perpendicular bisectors of the sides of the triangle
• The circumcircle's radius R is equal to a*b*c/(4*K), where a, b, and c are the lengths of the sides of the triangle, and K is its area.
• The largest triangle (by area) that can fit inside a circumcircle is an equilateral triangle.
• If the circumcircle's radius is 1, then the equilateral triangle's side length is sqrt(3) = 1.73.
• If the equilateral triangle's side length is 1, then the circumcircle's radius is R = 1/sqrt(3) = 0.58.
• The equilateral triangle occupies 41% of the circumcircle's area
• The tangents to the circumcircle at points A, B, and C intersect at three points, forming another triangle. This triangle is called the tangential triangle.

How it's related to the incircle:

• The incircle is always inside the circumcircle.
• The incenter I and the circumcenter O are separated by sqrt(R*(R-2r))

The radius of the circumcircle is:

```      abc
R  = -----                                       (1)
4K
```

How to find the circumcircle:

Example 1: An acute triangle has vertices A, B, and C at A = (1,3), B = (6,5), and C = (4,7). The first things we calculate are the lengths of the sides, a, b, and c, the semiperimeter s, and then the area of the triangle, K.

```A = (Ax, Ay) = ( 1, 3)
B = (Bx, By) = ( 6, 5)
C = (Cx, Cy) = ( 4, 7)

a = length of the side opposite vertex A
= length of the side BC
= sqrt((Cx-Bx)^2 + (Cy-By)^2)
= sqrt((4 - 6)^2 + (7 - 5)^2)
= sqrt(4 + 4) = sqrt(8)
= 2.8
b = length of the side opposite vertex B
= length of the side CA
= sqrt ((Ax-Cx)^2 + (Ay-Cy)^2)
= sqrt ((1 - 4)^2 + (3 - 7)^2)
= sqrt(9 + 16) = sqrt(25)
= 5
c = length of the side opposite vertex C
= length of the side AB
= sqrt ((Bx-Ax)^2 + (By-Ay)^2)
= sqrt ((6 - 1)^2 + (5 - 3)^2)
= sqrt(25 + 4) = sqrt(29)
= 5.4
s = (a+b+c)/2
= (2.8 + 5 + 5.4)/2
= 6.6
K = sqrt(s*(s-a)*(s-b)*(s-c))
= sqrt(6.6*(6.6-2.8)*(6.6-5)*(6.6-5.4))
= 7
```

The radius of the circumcircle is:

```     abc    2.8*5*5.4
R  = --- = -----------
4K      4*7

R  = 2.7
```

The method of finding the coordinates of the circumcenter O was described in circumcenter. Calculations are summarized:

```dA = 10
dB = 61
dC = 65
O = (Ox, Oy) = (3.4, 4.4)
```

The circumcenter is inside the triangle, as it must. Acute triangles' circumcenters are located inside the triangles, where as the circumcenters of obtuse triangles lie outside of them. The following example shows such a case.

Example 2: An obtuse triangle has vertices A = (2,5), B = (6,3), and C = (4,6). The radius of the circumcircle is R = 2.3, and the circumcenter O = (3.9, 3.8). If you sketch the triangle and circumcenter on grid paper, you can see that O lies outside the triangle.

```A = (Ax, Ay) = ( 2, 5)
B = (Bx, By) = ( 6, 3)
C = (Cx, Cy) = ( 4, 6)
a = 3.6
b = 2.2
c = 4.5
s = 5.2
K = 4
R = 2.3
O = (Ox, Oy) = (3.9, 3.8)
```

Compass and Straightedge Construction: The circumcenter is the point of intersection of two perpendicular bisectors. This fact allows you to find the circumcenter - and thus the circumcircle - in the same manner ancient geometers did: using just a compass and straightedge.

1. Construct a perpendicular bisector to side A
1. Place the compass pivot at one end of A (at the triangle vertex)
2. Open the compass so that its arc is longer than half of A
3. Draw a circle
4. Now move the compass pivot to the other end of A
5. Don't change the compass radius
6. Draw a second circle
7. The two circles intersect at two points. Call these A' and A''
8. Take a straightedge and draw a line connecting A' and A''
2. Construct a perpendicular bisector to side B
1. Use the same steps found in constructing the P.B. to side A
3. The point of intersection of the two perpendicular bisectors is the circumcenter
4. The radius R is the distance between O and any vertex
5. Construct the circumcircle
1. Construct a circle centered at O with radius R:
1. Place the compass pivot at O
2. Open the compass so that it touches any vertex
3. CHECK: The compass should be able to sweep through all three triangle vertices. If it doesn't, you may need to adjust either the compass pivot point or its radius
4. Draw a circle 360 degrees.
2. This is the circumcircle
3. You're done!

You can visualize this compass & straightedge construction very easily by viewing a short, silent, no-words movie here at MathOpenReference.com.

Point of Concurrency of Perpendicular Bisectors: The circumcenter is the point of concurrency of all three perpendicular bisectors.

Euclid showed that the perpendicular bisectors all meet at the same point for every possible triangle (which is the definition for the point of concurrency). He did this in Book 4, Proposition 5 of The Elements.

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.

I created a geometry-oriented spreadsheet in Excel that would graph three points, A, B, and C, the triangle, the circumcircle, and the circumcenter. Having to calculate real numbers forces one to get the formula exactly right. There is no "almost" in engineering and mathematics. A picture of the triangle and circumcircle is on my homenode, and will stay there for a brief time. If you read this node and would like to see a picture, please message me through E2 and I'll be glad to repost or to send you a copy.

Radial Angles around O The angles between vertices, as measured at the circumcenter, can be named φA, φB, and φC. Because they subtend the same arc of the circle as do the interior angles, they have twice the radial measure as do the interior angles. Thus, φA = 2 α, φB = 2 β, and φC = 2 γ The interior angles sum to 360, as the angles form a complete 2π radial arc.

Arc lengths The triangle cuts the circumcircle into three arc lengths. The arc lengths for AB, BC, and AB are, respectively, R φA, R φB, and R φC.

Angle of tangent lines at the vertices: The circumcircle intersects triangle ABC at the vertices A, B, and C. The tangent line makes angles with two of the sides. These angles are equal to the interior angles of the triangle, by the Alternate Segment Theorem. So for example, at vertex A, the two sides are AB and AC. These sides form angles beta and gamma at the base of the triangle (the base is side BC). These angles are easily calculatable if the sides are known by the law of cosines. Then the angle that side AB makes with the tangent line is equal to the opposite interior angle (the angle made by AC with base BC). This is angle β.

The Circumradius of Special Triangles The circumradius of the more common or interesting triangles is listed below.

``` Triangle                                   Inradius  Circumradius
==================================================================
3/ 4/ 5                                           1           5/2
5/12/13                                          30          13/2
8/15/17                                          60          19/2
7/24/25                                          84          25/2
20/21/29                                         210          29/2
Equilateral w side 1                       sqrt(3)/4     1/sqrt(3)
Equilateral w radius 1                             2             1
Isosceles Right w side 1               1/(2+sqrt(2))     1/sqrt(2)
Isosceles Right w hypotenuse 1     1/(2*(1+sqrt(2)))           1/2
30/60/90 w hypotenuse 1            1/(2*(1+sqrt(3)))           1/2
30/60/90 w short leg 1                 1/(3+sqrt(3))     1/sqrt(3)
30/60/90 w long leg 1                  1/(1+sqrt(3))             1
==================================================================
```

SOURCES:

Everything2 Writeups: Articles on (topic)

1. tongpoo, circumcenter, Dec. 2, 2001
2. IWhoSawTheFace, circumcenter, Nov. 7, 2011
3. tongpoo, triangle, Feb. 8, 2002

References: Useful books and references on geometry

1. H.S.M. Coxeter, Introduction to Geometry, 2nd Ed., (c) 1969
Next to Euclid's Elements, a true classic of geometry. Spare, elegant, beautiful.
§ 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. "Circumcircle" From MathWorld--A Wolfram Web Resource.
8. Weisstein, Eric W. "Incenter" From MathWorld--A Wolfram Web Resource.
9. Weisstein, Eric W. "Circumradius" From MathWorld--A Wolfram Web Resource.
10. Weisstein, Eric W. "Circumcenter" From MathWorld--A Wolfram Web Resource.
11. Weisstein, Eric W. "Tangential Triangle" From MathWorld--A Wolfram Web Resource.
12. Weisstein, Eric W. "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. It is a reference I use quite frequently, as the language is clean and to the point, without being unnecessarily over-mathematical, as Mathworld's information tends to be.

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