# circumradius

In geometry, the

A polygon which has a circumscribed circle is called a

A related notion is the one of a

All triangles are cyclic, i.e. every triangle has a circumscribed circle.

The circumcenter of a triangle can be found as the intersection of the three perpendicular bisectors. (A

In coastal navigation, 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.

The circumcentre's position depends on the type of triangle:

The diameter of the circumcircle can be computed as the length of any side of the triangle, divided by the sine of the opposite angle. (As a consequence of the law of sines, it doesn't matter which side is taken: the result will be the same.) 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 an infinitely large circle. Nearly collinear points often cause problems and errors in computation of the circumcircle.

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

Given

**circumscribed circle**or**circumcircle**of a polygon is a circle which passes through all the vertices of the polygon. The centre of this circle is called the**circumcenter**.A polygon which has a circumscribed circle is called a

**cyclic polygon**. All regular simple polygons, all triangles and all rectangles are cyclic.A related notion is the one of a

**minimum bounding circle**, which is the smallest circle that completely contains the polygon within it. Not every polygon has a circumscribed circle, as the vertices of a polygon do not need to all lie on a circle. Yet any polygon has a unique minimum bounding circle, which may be constructed by a linear time algorithm.^{[1]}Even if a polygon has a circumscribed circle, it may not coincide with its minimum bounding circle; for example, for an obtuse triangle, the minimum bounding circle has the hypotenuse as diameter and does not pass through the opposite vertex.## Circumcircles of triangles

All triangles are cyclic, i.e. every triangle has a circumscribed circle.

The circumcenter of a triangle can be found as the intersection of the three perpendicular bisectors. (A

*perpendicular bisector*is a line that forms a right angle with one of the triangle's sides and intersects that side at its midpoint.) This is because the circumcenter is equidistant from any pair of the triangle's points, and all points on the perpendicular bisectors are equidistant from those points of the triangle.In coastal navigation, 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.

The circumcentre's position depends on the type of triangle:

- If and only if a triangle is acute (all angles smaller than a right angle), the circumcenter lies inside the triangle
- If and only if it is obtuse (has one angle bigger than a right angle), the circumcenter lies outside
- If and only if it is a right triangle, the circumcenter lies on one of its sides (namely, the hypotenuse). This is one form of Thales' theorem.

The diameter of the circumcircle can be computed as the length of any side of the triangle, divided by the sine of the opposite angle. (As a consequence of the law of sines, it doesn't matter which side is taken: the result will be the same.) 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 an infinitely large circle. Nearly collinear points often cause problems and errors in computation of the circumcircle.

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

### Circumcircle equations

The circumcircle is given in Cartesian coordinates by the equation*A*,*B*and*C*are the vertices of the triangle, and the solution for**v**is the circumcircle. (Note**A**^{2}=*A*_{x}^{2}+*A*_{y}^{2}.)Given

- , , ,

**v**^{2}− 2**Sv**−*b*= 0 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 S at infinity), (**v**−**S**/*a*)^{2 = b/a + S2/a2, giving the circumcenter S/a and the circumradius √ (b/a + S2/a2). This approach should also work for the circumsphere of a tetrahedron. An equation for the circumcircle in trilinear coordinates x : y : z is a/x + b/y + c/z = 0. An equation for the circumcircle in barycentric coordinates x : y : z is 1/x + 1/y + 1/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. Coordinates of circumcenter The circumcenter has trilinear coordinates (cos , cos , cos ) where are the angles of the triangle. The circumcenter has barycentric coordinates where are edge lengths ( respectively) of the triangle. The angles at which the circle meets the sides The angles at which the circumscribed circle meet the sides of the triangle coincide with angles at which sides meet each other. The side opposite angle α meets the circle twice: once at each end; in each case at angle α (similarly for the other two angles). Triangle centers on the circumcircle of triangle ABC In this section, the vertex angles are labeled A, B, C and all coordinates are trilinear coordinates: Steiner point = bc/ (b2 − c2) : ca/ (c2 − a2) : ab/(a2 − b2) = the nonvertex point of intersection of the circumcircle with the Steiner ellipse. (The Steiner ellipse, with center = centroid(ABC), is the ellipse of least area that passes through A, B, and C. An equation for this ellipse is 1/(ax) + 1/(by) + 1/(cz) = 0.) Tarry point = sec (A + ω) : sec (B + ω) : sec (C + ω) = antipode of the Steiner point Focus of the Kiepert parabola = csc (B − C) : csc (C − A) : csc (A − B) Cyclic quadrilaterals Cyclic quadrilaterals Main article: Cyclic quadrilateral Quadrilaterals that can be circumscribed have particular properties including the fact that opposite angles are supplementary angles (adding up to 180° or π radians). See also inscribed circle Jung's theorem, an inequality relating the diameter of a point set to the radius of its minimum bounding circle References ^ Megiddo, N. (1983). "Linear-time algorithms for linear programming in R3 and related problems". SIAM Journal on Computing 12: 759–776. Kimberling, Clark (1998). "Triangle centers and central triangles". Congressus Numerantium 129: i–xxv, 1–295. External links Triangle centers by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas. Triangle circumcircle and circumcenter With interactive animation Circumcircle at MathWorld Steiner circumellipse at MathWorld An interactive Java applet for the circumcenter Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences...... Click the link for more information. POLYGONE is an Electronic Warfare Tactics Range located on the border between France and Germany. It is one of only two in Europe, the other being RAF Spadeadam.The range, also referred to as the Multi-national Aircrew Electronic Warfare Tactics Facility (MAEWTF), is..... Click the link for more information. In geometry, the centre (or center, in American English) of an object is a point in some sense in the middle of the object. If geometry is regarded as the study of isometry groups then the centre is a fixed point of the isometries...... Click the link for more information. simple polygon is a polygon whose sides do not intersect. They are also called Jordan polygons, because the Jordan curve theorem can be used to prove that such a polygon divides the plane into two regions, the region inside it and the region outside it...... Click the link for more information. A triangle is one of the basic shapes of geometry: a polygon with three corners or and three sides or edges which are straight line segments.In Euclidean geometry any three non-collinear points determine a triangle and a unique plane, i.e...... Click the link for more information. rectangle is defined as a quadrilateral where all four of its angles are right angles.From this definition, it follows that a rectangle has two pairs of parallel sides; that is, a rectangle is a parallelogram...... Click the link for more information. In computational complexity theory, an algorithm is said to take linear time, or O(n) time, if the asymptotic upper bound for the time it requires is proportional to the size of the input, which is usually denoted n...... Click the link for more information. A triangle is one of the basic shapes of geometry: a polygon with three corners or and three sides or edges which are straight line segments.In Euclidean geometry any three non-collinear points determine a triangle and a unique plane, i.e...... Click the link for more information. perpendicular (or orthogonal) to each other if they form congruent adjacent angles. The term may be used as a noun or adjective. Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B...... Click the link for more information. Bisection is the general activity of dividing something into two parts.Bisect may mean: bisection, in geometry, dividing something into two equal parts bisect (philately), the use of postage stamp halves ..... Click the link for more information. Pilotage is the use of fixed visual references on the ground or sea by means of sight or radar to guide oneself to a destination, sometimes with the help of a map or nautical chart. People use pilotage for activities such as guiding vessels and aircraft, hiking and Scuba diving...... Click the link for more information. A position line is a line that can be identified both on a nautical chart or aeronautical chart and by observation out on the surface of the earth. The intersection of two position lines is a fix that used in position fixing to identify the navigator's location...... Click the link for more information. This article is about the Sextant as used for navigation. For the astronomer's sextant, see Sextant (astronomical). For the history and development of the sextant see Reflecting instrumentsA sextant..... Click the link for more information. For other uses, see Compass (disambiguation).COMPASS is an acronym for COMPrehensive ASSembler. COMPASS is a macro assembly language on Control Data Corporation's 3000 series, and on the 60-bit CDC 6000 series, 7600 and..... Click the link for more information. “Iff” redirects here. For other uses, see IFF. If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a logical connective between statements which means that the truth of either one of the statements..... Click the link for more information. hypotenuse of a right triangle is the triangle's longest side; the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares..... Click the link for more information. Thales' theorem (named after Thales of Miletus) states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle...... Click the link for more information. diameter (Greek words diairo = divide and metro = measure) of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...... Click the link for more information. trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications...... Click the link for more information. angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept..... Click the link for more information. law of sines (or sine law, sine formula) is a statement about arbitrary triangles in the plane. If the sides of the triangle are a, b and c and the angles opposite to those sides are A, B and C..... Click the link for more information. In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant points, six lying on the triangle itself (unless the triangle is obtuse)...... Click the link for more information. In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of ...... Click the link for more information. In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side or an extension of the opposite side...... Click the link for more information. Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral; it passes through several important points determined from the triangle. In the image, the Euler line is shown in red...... Click the link for more information. In geometry, the isogonal conjugate of a point P with respect to a triangle ABC is constructed by reflecting the lines PA, PB, and PC about the angle bisectors of A, B, and C...... Click the link for more information. In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side or an extension of the opposite side...... Click the link for more information. bounding sphere, enclosing sphere or enclosing ball for that set is an n-dimensional solid sphere containing all of these objects.In the plane the terms bounding or enclosing circle are used...... Click the link for more information. line can be described as an ideal zero-width, infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves") containing an infinite number of points. In Euclidean geometry, exactly one line can be found that passes through any two points...... Click the link for more information. In mathematics, and computational geometry, a Delaunay triangulation or Delone triangularization for a set P of points in the plane is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P)...... Click the link for more information. 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