Circumcircle

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 equidistant from all vertices, and can thus be constructed as the point of intersection between any two perpendicular bisectors of the triangle's sides. It 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 triangles, rectangles, isosceles trapezoids, right kites, and regular polygons are cyclic, but not every polygon is.

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. These locational features can be seen by considering the trilinear or barycentric coordinates given below 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.

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.

Circumcircle equations

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 \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 \mathbf v = (v_x,v_y) in the Cartesian plane satisfying the equations \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 \mathbf u of the circle. Using the polarization identity, these equations reduce to the condition that the matrix \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: \det\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 \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|\mathbf v|^2 - 2\mathbf{Sv} - b = 0 where \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) – \left|\mathbf v - \tfrac{\mathbf S}{a}\right|^2 = \tfrac{b}{a} + \tfrac . 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-handed) sense about is the following: \mathrm{R} (s) = \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].

Trilinear and barycentric coordinates

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

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: \begin{align} \mathbf{a} &= \mathbf{A}-\mathbf{C}, \\ \mathbf{b} &= \mathbf{B}-\mathbf{C}. \end{align} The circumradius is then r = \frac {\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\left\|\mathbf{a} - \mathbf{b}\right\|} {2 \left\|\mathbf{a}\times\mathbf{b}\right\|} p_0 = \frac {\left(\left\|\mathbf{a}\right\|^2\mathbf{b} - \left\|\mathbf{b}\right\|^2\mathbf{a}\right) \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: \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 = \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 = \frac {\left(\left(\left\|\mathbf{a}\right\|^2\mathbf{b} - \left\|\mathbf{b}\right\|^2\mathbf{a}\right) \cdot \mathbf{b}\right) \mathbf{a} - \left(\left(\left\|\mathbf{a}\right\|^2\mathbf{b} - \left\|\mathbf{b}\right\|^2\mathbf{a}\right) \cdot \mathbf{a}\right) \mathbf{b}} {2 \left(\left\|\mathbf{a}\right\|^{2}\left\|\mathbf{b}\right\|^2 - (\mathbf{a} \cdot \mathbf{b})^2\right)} + \mathbf{C} which can be simplified to: p_0 = \frac {\left\|\mathbf{a}\right\|^{2}\left\|\mathbf{b}\right\|^{2}(\mathbf{a} + \mathbf{b}) - (\mathbf{a}\cdot\mathbf{b}) \left(\left\|\mathbf{a}\right\|^{2}\mathbf{b} + \left\|\mathbf{b}\right\|^{2}\mathbf{a}\right)} {2 \left(\left\|\mathbf{a}\right\|^{2}\left\|\mathbf{b}\right\|^2 - (\mathbf{a} \cdot \mathbf{b})^2\right)} + \mathbf{C} The Cartesian coordinates of the circumcenter U = \left(U_x, U_y\right) are \begin{align} U_x &= \frac{1}{D}\left[\left(A_x^2 + A_y^2\right)(B_y - C_y) + \left(B_x^2 + B_y^2\right)(C_y - A_y) + \left(C_x^2 + C_y^2\right)(A_y - B_y)\right] \\[5pt] U_y &= \frac{1}{D}\left[\left(A_x^2 + A_y^2\right)(C_x - B_x) + \left(B_x^2 + B_y^2\right)(A_x - C_x) + \left(C_x^2 + C_y^2\right)(B_x - A_x)\right] \end{align} with D = 2\left[A_x(B_y - C_y) + B_x(C_y - A_y) + C_x(A_y - B_y)\right].\, 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' = A-A = (A'_x,A'_y) = (0,0). In this case, the coordinates of the vertices B'=B-A and C'=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' = (U'_x, U'_y) of the triangle follow as \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' = 2(B'_x C'_y - B'_y C'_x). \, Due to the translation of vertex to the origin, the circumradius can be computed as r = \|U'\| = \sqrt{{U'_x}^2 + {U'_y}^2} and the actual circumcenter of follows as U = U' + A \cos \alpha : \cos \beta : \cos \gamma where are the angles of the triangle. In terms of the side lengths , the trilinears are a\left(b^2 + c^2 - a^2\right) : b\left(c^2 + a^2 - b^2\right) : c\left(a^2 + b^2 - c^2\right). a^2\left(b^2 + c^2 - a^2\right):\; b^2\left(c^2 + a^2 - b^2\right):\; c^2\left(a^2 + b^2 - c^2\right),\, where are edge lengths respectively) of the triangle. In terms of the triangle's angles , the barycentric coordinates of the circumcenter are \sin 2\alpha :\sin 2\beta :\sin 2\gamma . U = \frac {a^2\left(b^2 + c^2 - a^2\right)A + b^2\left(c^2 + a^2 - b^2\right)B + c^2\left(a^2 + b^2 - c^2\right)C} {a^2\left(b^2 + c^2 - a^2\right) + b^2\left(c^2 + a^2 - b^2\right) + c^2\left(a^2 + b^2 - c^2\right)}. Here is the vector of the circumcenter and are the vertex vectors. The divisor here equals where is the area of the triangle. \mathrm{P_1} = \begin{bmatrix} x_1 \\ y_1 \\ z_1 \end{bmatrix}, \mathrm{P_2} = \begin{bmatrix} x_2 \\ y_2 \\ z_2 \end{bmatrix}, \mathrm{P_3} = \begin{bmatrix} x_3 \\ y_3 \\ z_3 \end{bmatrix} Then the radius of the circle is given by \mathrm{r} = \frac {\left|P_1 - P_2\right| \left|P_2 - P_3\right|\left|P_3 - P_1\right|} {2 \left|\left(P_1 - P_2\right) \times \left(P_2 - P_3\right)\right|} The center of the circle is given by the linear combination \mathrm{P_c} = \alpha \, P_1 + \beta \, P_2 + \gamma \, P_3 where \begin{align} \alpha = \frac {\left|P_2 - P_3\right|^2 \left(P_1 - P_2\right) \cdot \left(P_1 - P_3\right)} {2 \left|\left(P_1 - P_2\right) \times \left(P_2 - P_3\right)\right|^2} \\ \beta = \frac {\left|P_1 - P_3\right|^2 \left(P_2 - P_1\right) \cdot \left(P_2 - P_3\right)} {2 \left|\left(P_1 - P_2\right) \times \left(P_2 - P_3\right)\right|^2} \\ \gamma = \frac {\left|P_1 - P_2\right|^2 \left(P_3 - P_1\right) \cdot \left(P_3 - P_2\right)} {2 \left|\left(P_1 - P_2\right) \times \left(P_2 - P_3\right)\right|^2} \end{align} Steiner point: the non-vertex point of intersection of the circumcircle with the Steiner ellipse. \frac{bc}{b^2 - c^2} : \frac{ca}{c^2 - a^2} : \frac{ab}{a^2 - b^2} (The Steiner ellipse, with center = centroid ( ), is the ellipse of least area that passes through . An equation for this ellipse is Tarry point: antipode of the Steiner point \sec(A + \omega) : \sec(B + \omega) : \sec(C + \omega) Focus of the Kiepert parabola: \csc(B-C) : \csc(C-A) : \csc(A-B). \text{diameter} = \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}. As a consequence of the law of sines, it does not matter which side and opposite angle are taken: the result will be the same. The diameter of the circumcircle can also be expressed as \begin{align} \text{diameter} & {}= \frac{abc}{2\cdot\text{area}} = \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=\tfrac{a+b+c}{2} is the semiperimeter. The expression \scriptstyle \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 \text{diameter} = \sqrt{\frac{2 \cdot \text{area}}{\sin A \sin B \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 \overline{OI} = \sqrt{R(R - 2r)}, where is the incircle radius and is the circumcircle radius; hence the circumradius is at least twice the inradius (Euler's triangle inequality), with equality only in the equilateral case. The distance between and the orthocenter is \overline{OH} = \sqrt{R^2 - 8R^2\cos A \cos B \cos C} = \sqrt{9R^2 - \left(a^2 + b^2 + c^2\right)}. For centroid and nine-point center we have \begin{align} \overline{IG} & The product of the incircle radius and the circumcircle radius of a triangle with sides is rR = \frac{abc}{2(a + b + c)}. With circumradius , sides , and medians , we have \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 4R^2 h^2\left(t^2 - h^2\right) = t^4\left(m^2 - h^2\right). 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 \overline{OI}=\sqrt{R(R-2r)}.

References

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

Text derived from https://en.wikipedia.org/w/index.php?oldid=1375949773 under CC BY-SA 4.0; modifications have been made.

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