Circumcircle

 ( Circles Defined For A Triangle ) 

In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcenter is the point of intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center.

More generally, an -sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon, or in the special case , a cyclic quadrilateral. All , isosceles trapezoids, , and regular polygons are cyclic, but not every polygon is.

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Straightedge and compass construction The circumcircle of a triangle can be constructed using straightedge and compass by first constructing any two of the three perpendicular bisectors of the sides; their point of intersection is the circumcenter. The circumcircle can immediately be drawn as the circle centered there and passing through one of the triangle's vertices; its radius is the circumradius.

Any point on a perpendicular bisector of one side is equidistant from the two adjacent vertices of the triangle. Therefore any point which is simultaneously on two of the perpendicular bisectors must be equidistant from all three vertices.

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Alternative construction An alternative method to determine the circumcenter is to draw any two lines each one departing from one of the vertices at an angle with the common side, the common angle of departure being 90° minus the angle of the opposite vertex. (In the case of the opposite angle being obtuse, drawing a line at a negative angle means going outside the triangle.)

In pilotage, a triangle's circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies.

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Location relative to the triangle The circumcenter's position depends on the type of triangle:

• For an acute triangle (all angles smaller than a right angle), the circumcenter always lies inside the triangle.
• For a right triangle, the circumcenter always lies at the midpoint of the hypotenuse. This is one form of Thales' theorem.
• For an obtuse triangle (a triangle with one angle bigger than a right angle), the circumcenter always lies outside the triangle.

These locational features can be seen by considering the trilinear or barycentric coordinates given above for the circumcenter: all three coordinates are positive for any interior point, at least one coordinate is negative for any exterior point, and one coordinate is zero and two are positive for a non-vertex point on a side of the triangle.

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Angles

The angles which the circumscribed circle forms with the sides of the triangle coincide with angles at which sides meet each other. The side opposite angle meets the circle twice: once at each end; in each case at angle (similarly for the other two angles). This is due to the alternate segment theorem, which states that the angle between the tangent and chord equals the angle in the alternate segment.

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Circumcircle equations

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Cartesian coordinates In the Euclidean plane, it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that

$\backslash begin\{align\}$

 \mathbf{A} &= (A_x, A_y) \\
 \mathbf{B} &= (B_x, B_y) \\
 \mathbf{C} &= (C_x, C_y)
     

\end{align}

are the coordinates of points . The circumcircle is then the locus of points $\backslash mathbf\; v\; =\; (v\_x,v\_y)$ in the Cartesian plane satisfying the equations

$\backslash begin\{align\}$

 |\mathbf{v} - \mathbf{u}|^2 &= r^2 \\
 |\mathbf{A} - \mathbf{u}|^2 &= r^2 \\
 |\mathbf{B} - \mathbf{u}|^2 &= r^2 \\
 |\mathbf{C} - \mathbf{u}|^2 &= r^2
     

\end{align}

guaranteeing that the points are all the same distance from the common center $\backslash mathbf\; u$ of the circle. Using the polarization identity, these equations reduce to the condition that the matrix

$\backslash begin\{bmatrix\}$

 |\mathbf{v}|^2 & -2v_x & -2v_y & -1 \\
 |\mathbf{A}|^2 & -2A_x & -2A_y & -1 \\
 |\mathbf{B}|^2 & -2B_x & -2B_y & -1 \\
 |\mathbf{C}|^2 & -2C_x & -2C_y & -1
     

\end{bmatrix}

has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix:

$\backslash det\backslash begin\{bmatrix\}$

 |\mathbf{v}|^2 & v_x & v_y & 1 \\
 |\mathbf{A}|^2 & A_x & A_y & 1 \\
 |\mathbf{B}|^2 & B_x & B_y & 1 \\
 |\mathbf{C}|^2 & C_x & C_y & 1
     

\end{bmatrix}=0.

Using cofactor expansion, let

$\backslash begin\{align\}$

 S_x &= \frac{1}{2}\det\begin{bmatrix}
   |\mathbf{A}|^2 & A_y & 1 \\
   |\mathbf{B}|^2 & B_y & 1 \\
   |\mathbf{C}|^2 & C_y & 1
 \end{bmatrix}, \\[5pt]
 S_y &= \frac{1}{2}\det\begin{bmatrix}
   A_x & |\mathbf{A}|^2 & 1 \\
   B_x & |\mathbf{B}|^2 & 1 \\
   C_x & |\mathbf{C}|^2 & 1
 \end{bmatrix}, \\[5pt]
 a &= \det\begin{bmatrix}
   A_x & A_y & 1 \\
   B_x & B_y & 1 \\
   C_x & C_y & 1
 \end{bmatrix}, \\[5pt]
 b &= \det\begin{bmatrix}
    A_x & A_y & |\mathbf{A}|^2 \\
    B_x & B_y & |\mathbf{B}|^2 \\
    C_x & C_y & |\mathbf{C}|^2
 \end{bmatrix}
     

\end{align}

we then have $a|\backslash mathbf\; v|^2\; -\; 2\backslash mathbf\{Sv\}\; -\; b\; =\; 0$ where $\backslash mathbf\; S\; =\; (S\_x,\; S\_y),$ and – assuming the three points were not in a line (otherwise the circumcircle is that line that can also be seen as a generalized circle with at infinity) – $\backslash left|\backslash mathbf\; v\; -\; \backslash tfrac\{\backslash mathbf\; S\}\{a\}\backslash right|^2\; =\; \backslash tfrac\{b\}\{a\}\; +\; \backslash tfrac$

giving the circumcenter $\backslash tfrac\{\backslash mathbf\; S\}\{a\}$ and the circumradius $\backslash sqrt\{\backslash tfrac\{b\}\{a\}\; +\; \backslash tfrac\{|\backslash mathbf\; S|^2\}\{a^2\}\}.$ A similar approach allows one to deduce the equation of the circumsphere of a tetrahedron.>

Parametric equation A unit vector perpendicular to the plane containing the circle is given by $\backslash widehat\{n\}\; =\; \backslash frac\{(P\_2\; -\; P\_1)\; \backslash times\; (P\_3\; -\; P\_1)\}\{$

.

Hence, given the radius, , center, , a point on the circle, and a unit normal of the plane containing the circle, one parametric equation of the circle starting from the point and proceeding in a positively oriented (i.e., right-hand rule) sense about is the following:

$\backslash mathrm\{R\}\; (s)\; =\; \backslash mathrm\{P\_c\}\; +$

 \cos\left(\frac{\mathrm{s}}{\mathrm{r}}\right)
   (P_0 - P_c) +
 \sin\left(\frac{\mathrm{s}}{\mathrm{r}}\right)
   \left[\widehat{n} \times(P_0 - P_c)\right].
     

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Trilinear and barycentric coordinates An equation for the circumcircle in trilinear coordinates is $\backslash tfrac\{a\}\{x\}\; +\; \backslash tfrac\{b\}\{y\}\; +\; \backslash tfrac\{c\}\{z\}\; =0.$ An equation for the circumcircle in barycentric coordinates is $\backslash tfrac\{a^2\}\{x\}\; +\; \backslash tfrac\{b^2\}\{y\}\; +\; \backslash tfrac\{c^2\}\{z\}\; =0.$

The isogonal conjugate of the circumcircle is the line at infinity, given in trilinear coordinates by $ax+by+cz=0$ and in barycentric coordinates by $x+y+z=0.$

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Higher dimensions Additionally, the circumcircle of a triangle embedded in three dimensions can be found using a generalized method. Let be three-dimensional points, which form the vertices of a triangle. We start by transposing the system to place at the origin:

$\backslash begin\{align\}$

 \mathbf{a} &= \mathbf{A}-\mathbf{C}, \\
 \mathbf{b} &= \mathbf{B}-\mathbf{C}.
     

\end{align}

The circumradius is then

$r\; =\; \backslash frac$

            {\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\left\|\mathbf{a} - \mathbf{b}\right\|}
            {2 \left\|\mathbf{a}\times\mathbf{b}\right\|}
        = \frac{\left\|\mathbf{a} - \mathbf{b}\right\|}{2 \sin\theta}
        = \frac{\left\|\mathbf{A} - \mathbf{B}\right\|}{2 \sin\theta},
     

where is the interior angle between and . The circumcenter, , is given by

$p\_0\; =\; \backslash frac\{(\backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |^2\backslash mathbf\{b\}\; -\; \backslash left\backslash |\backslash mathbf\{b\}\backslash right\backslash |^2\backslash mathbf\{a\})$

                     \times (\mathbf{a} \times \mathbf{b})}
                 {2 \left\|\mathbf{a}\times\mathbf{b}\right\|^2} + \mathbf{C}.
     

This formula only works in three dimensions as the cross product is not defined in other dimensions, but it can be generalized to the other dimensions by replacing the cross products with following identities:

$\backslash begin\{align\}$

 \mathbf{u} \times (\mathbf{v} \times \mathbf{w}) &= (\mathbf{u} \cdot \mathbf{w})\mathbf{v} - (\mathbf{u} \cdot \mathbf{v})\mathbf{w}, \\
   \left\|\mathbf{u} \times \mathbf{v}\right\|^2  &= \left\|\mathbf{u}\right\|^2 \left\|\mathbf{v}\right\|^2 - (\mathbf{u} \cdot \mathbf{v})^2.
     

\end{align}

This gives us the following equation for the circumradius :

$r\; =\; \backslash frac$

            {\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\left\|\mathbf{a} - \mathbf{b}\right\|}
            {2 \sqrt{\left\|\mathbf{a}\right\|^{2}\left\|\mathbf{b}\right\|^2 - (\mathbf{a} \cdot \mathbf{b})^2}}
     

and the following equation for the circumcenter :

$p\_0\; =\; \backslash frac\{((\backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |^2\backslash mathbf\{b\}\; -\; \backslash left\backslash |\backslash mathbf\{b\}\backslash right\backslash |^2\backslash mathbf\{a\})$

                    \cdot \mathbf{b}) \mathbf{a} -
                  ((\left\|\mathbf{a}\right\|^2\mathbf{b} - \left\|\mathbf{b}\right\|^2\mathbf{a})
                    \cdot \mathbf{a}) \mathbf{b}}
                 {2 (\left\|\mathbf{a}\right\|^{2}\left\|\mathbf{b}\right\|^2 - (\mathbf{a} \cdot \mathbf{b})^2)}
          + \mathbf{C}
     

which can be simplified to:

$p\_0\; =\; \backslash frac\{\backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |^\{2\}\backslash left\backslash |\backslash mathbf\{b\}\backslash right\backslash |^\{2\}(\backslash mathbf\{a\}\; +\; \backslash mathbf\{b\})$

                  - (\mathbf{a}\cdot\mathbf{b})(\left\|\mathbf{a}\right\|^{2}\mathbf{b} + \left\|\mathbf{b}\right\|^{2}\mathbf{a})}
                 {2 (\left\|\mathbf{a}\right\|^{2}\left\|\mathbf{b}\right\|^2 - (\mathbf{a} \cdot \mathbf{b})^2)}
          + \mathbf{C}
     

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Circumcenter coordinates

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Cartesian coordinates The Cartesian coordinates of the circumcenter $U\; =\; \backslash left(U\_x,\; U\_y\backslash right)$ are

$\backslash begin\{align\}$

 U_x &= \frac{1}{D}\left[(A_x^2 + A_y^2)(B_y - C_y) + (B_x^2 + B_y^2)(C_y - A_y) + (C_x^2 + C_y^2)(A_y - B_y)\right] \\[5pt]
 U_y &= \frac{1}{D}\left[(A_x^2 + A_y^2)(C_x - B_x) + (B_x^2 + B_y^2)(A_x - C_x) + (C_x^2 + C_y^2)(B_x - A_x)\right]
     

\end{align}

with

$D\; =\; 2\backslash leftA\_x(B\_y.\backslash ,$

Without loss of generality this can be expressed in a simplified form after translation of the vertex to the origin of the Cartesian coordinate systems, i.e., when $A\text{'}\; =\; A-A\; =\; (A\text{'}\_x,A\text{'}\_y)\; =\; (0,0).$ In this case, the coordinates of the vertices $B\text{'}=B-A$ and $C\text{'}=C-A$ represent the vectors from vertex to these vertices. Observe that this trivial translation is possible for all triangles, and the coordinates of the circumcenter $U\text{'}\; =\; (U\text{'}\_x,\; U\text{'}\_y)$ of the triangle follow as

$\backslash begin\{align\}$

 U'_x &= \frac{1}{D'}\left[C'_y({B'_x}^2 + {B'_y}^2) - B'_y({C'_x}^2 + {C'_y}^2)\right], \\[5pt]
 U'_y &= \frac{1}{D'}\left[B'_x({C'_x}^2 + {C'_y}^2) - C'_x({B'_x}^2 + {B'_y}^2)\right]
     

\end{align}

with

$D\text{'}\; =\; 2(B\text{'}\_x\; C\text{'}\_y\; -\; B\text{'}\_y\; C\text{'}\_x).\; \backslash ,$

Due to the translation of vertex to the origin, the circumradius can be computed as

$r\; =\; \backslash |U\text{'}\backslash |\; =\; \backslash sqrt\; =\; \backslash frac${2|\Delta ABC|} \\5pt

                 & {}= \frac{abc}{2\sqrt{s(s - a)(s - b)(s - c)}}\\[5pt]
                 & {}= \frac{2abc}{\sqrt{(a + b + c)(-a + b + c)(a - b + c)(a + b - c)}}
     

\end{align}

where are the lengths of the sides of the triangle and $s=\backslash tfrac\{a+b+c\}\{2\}$ is the semiperimeter. The expression $\backslash scriptstyle\; \backslash sqrt\{s(s-a)(s-b)(s-c)\}$ above is the area of the triangle, by Heron's formula.

Trigonometric expressions for the diameter of the circumcircle include

$\backslash text\{diameter\}\; =\; \backslash sqrt\{\backslash frac\{2\; \backslash cdot\; \backslash text\{area\}\}\{\backslash sin\; A\; \backslash sin\; B\; \backslash sin\; C\}\}.$

The triangle's nine-point circle has half the diameter of the circumcircle.

In any given triangle, the circumcenter is always collinear with the centroid and orthocenter. The line that passes through all of them is known as the Euler line.

The isogonal conjugate of the circumcenter is the orthocenter.

The useful minimum bounding circle of three points is defined either by the circumcircle (where three points are on the minimum bounding circle) or by the two points of the longest side of the triangle (where the two points define a diameter of the circle). It is common to confuse the minimum bounding circle with the circumcircle.

The circumcircle of three collinear points is the line on which the three points lie, often referred to as a circle of infinite radius. Nearly collinear points often lead to numerical instability in computation of the circumcircle.

Circumcircles of triangles have an intimate relationship with the Delaunay triangulation of a set of points.

By Euler's theorem in geometry, the distance between the circumcenter and the incenter is

$\backslash overline\{OI\}\; =\; \backslash sqrt\{R(R\; -\; 2r)\},$

where is the incircle radius and is the circumcircle radius; hence the circumradius is at least twice the inradius (Euler inequality), with equality only in the equilateral case.Nelson, Roger, "Euler's triangle inequality via proof without words," Mathematics Magazine 81(1), February 2008, 58-61. See in particular p. 198.

The distance between and the orthocenter is See in particular p. 449.

$\backslash overline\{OH\}\; =\; \backslash sqrt\{R^2\; -\; 8R^2\backslash cos\; A\; \backslash cos\; B\; \backslash cos\; C\}\; =\; \backslash sqrt\{9R^2\; -\; (a^2\; +\; b^2\; +\; c^2)\}.$

For centroid and nine-point center we have

$\backslash begin\{align\}$

\overline{IG} &< \overline{IO}, \\ 2\overline{IN} &< \overline{IO}, \\ \overline{OI}^2 &= 2R\cdot \overline{IN}. \end{align}

The product of the incircle radius and the circumcircle radius of a triangle with sides is Republished by Dover Publications as Advanced Euclidean Geometry, 1960 and 2007.

$rR\; =\; \backslash frac\{abc\}\{2(a\; +\; b\; +\; c)\}.$

With circumradius , sides , and medians , we have

$\backslash begin\{align\}$

      3\sqrt{3}R &\geq a + b + c \\[5pt]
            9R^2 &\geq a^2 + b^2 + c^2 \\[5pt]
 \frac{27}{4}R^2 &\geq m_a^2 + m_b^2 + m_c^2.
     

\end{align}

If median , altitude , and internal bisector all emanate from the same vertex of a triangle with circumradius , then Reprinted by Dover Publications, 2007.

$4R^2\; h^2(t^2\; -\; h^2)\; =\; t^4(m^2\; -\; h^2).$

Carnot's theorem states that the sum of the distances from the circumcenter to the three sides equals the sum of the circumradius and the inradius. Here a segment's length is considered to be negative if and only if the segment lies entirely outside the triangle.

If a triangle has two particular circles as its circumcircle and incircle, there exist an infinite number of other triangles with the same circumcircle and incircle, with any point on the circumcircle as a vertex. (This is the case of Poncelet's porism). A necessary and sufficient condition for such triangles to exist is the above equality $\backslash overline\{OI\}=\backslash sqrt\{R(R-2r)\}.$

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Cyclic polygons A set of points lying on the same circle are called concyclic points, and a polygon whose vertices are concyclic is called a cyclic polygon. Every triangle is concyclic, but polygons with more than three sides are not in general.

Cyclic polygons, especially four-sided cyclic quadrilaterals, have various special properties. In particular, the opposite angles of a cyclic quadrilateral are supplementary angles (adding up to 180° or π radians).

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See also

• Circumcenter of mass
• Circumscribed sphere
• Circumcevian triangle
• Inscribed circle
• Kosnita theorem
• Lester's theorem
• Problem of Apollonius

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External links

• Derivation of formula for radius of circumcircle of triangle at Mathalino.com
• Semi-regular angle-gons and side-gons: respective generalizations of rectangles and rhombi at Dynamic Geometry Sketches, interactive dynamic geometry sketch.
• Weisstein, Eric W. "Circumcircle", "Cyclic Polygon". MathWorld.
• Triangle circumcircle and circumcenter With interactive animation
• An interactive Java applet for the circumcenter

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