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, but every polygon has a unique minimum bounding circle, which may be constructed by a linear time algorithm. 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 longest side as diameter and does not pass through the opposite vertex.
Circumscribed circles of triangles
All triangles are cyclic, i.e. every triangle has a circumscribed circle.[nb 1]
The circumcenter of a triangle can be found as the intersection of any two 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 vertices, and all points on the perpendicular bisectors are equidistant from two of the vertices of the triangle.
Alternate method to determine the circumcenter: draw any two lines departing the vertices at an angle with the common side, equal to 90 degrees minus the angle of the opposite vertex.
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 circumcenter'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 at the center of 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 does not matter which side is taken: the result will be the same.) The triangle's nine-point circle has half the diameter of the circumcircle. The diameter of the circumcircle of the triangle ΔABC is
where a, b, c are the lengths of the sides of the triangle and s = (a + b + c)/2 is the semiperimeter. The expression above is the area of the triangle, by Heron's formula. Trigometric expressions for the diameter of the circumcircle include:p.379
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.
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. Thus suppose that
are the coordinates of points A, B, and C. The circumcircle is then the locus of points v = (vx,vy) in the Cartesian plane satisfying the equations
guaranteeing that the points A, B, C, and v are all the same distance r from the common center u of the circle. Using the polarization identity, these equations reduce to the condition that the matrix
Expanding by cofactor expansion, let
we then have a|v|2 − 2Sv − 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 + |S|2/a2, giving the circumcenter S/a and the circumradius √ (b/a + |S|2/a2). A similar approach allows one to deduce the equation of the circumsphere of a tetrahedron.
Additionally, the circumcircle of a triangle embedded in d dimensions can be found using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle. We start by transposing the system to place C at the origin:
Case of three dimensions:
The circumradius, r, is then
where θ is the interior angle between a and b. The circumcenter, p0, is given by
These formulas can be used directly in the three dimensions only because the cross product is not defined as binary operation otherwise, but these formulas can be easy extended to the d dimensions case using following identities valid for arbitrary vectors in three dimensions:
Coordinates of a triangle's circumcenter
The Cartesian coordinates of the circumcenter are
Without loss of generality this can be expressed in a simplified form after translation of the vertex A 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 A' to these vertices. Observe that this trivial translation is possible for all triangles and the circumcenter coordinates of the triangle A'B'C' follow as
Barycentric coordinates as a function of the side lengths
where a, b, c are edge lengths (BC, CA, AB respectively) of the triangle.
Barycentric coordinates from cross- and dot-products
In Euclidean space, there is a unique circle passing through any given three non-collinear points P1, P2, and P3. Using Cartesian coordinates to represent these points as spatial vectors, it is possible to use the dot product and cross product to calculate the radius and center of the circle. Let
Then the radius of the circle is given by
The center of the circle is given by the linear combination
Parametric equation of a triangle's circumcircle
Hence, given the radius, r, center, Pc, a point on the circle, P0 and a unit normal of the plane containing the circle, , one parametric equation of the circle starting from the point P0 and proceeding in a positively oriented (i.e., right-handed) sense about is the following:
The angles which the circle forms with a triangle's sides
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 α meets the circle twice: once at each end; in each case at angle α (similarly for the other two angles). The alternate segment theorem states that the angle between the tangent and chord equals the angle in the alternate segment.
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).
Other properties of the circumcircle of a triangle
The circumcircle radius is no smaller than twice the incircle radius (Euler's triangle inequality).
The distance between the circumcenter and the incenter is where r is the incircle radius and R is the circumcircle radius.
The product of the incircle radius and the circumcircle radius of a triangle with sides a, b, and c is
Quadrilaterals that can be circumscribed have particular properties including the fact that opposite angles are supplementary angles (adding up to 180° or π radians).
For a cyclic polygon with an odd number of sides, all angles are equal if and only if the polygon is regular. A cyclic polygon with an even number of sides has all angles equal if and only if the alternate sides are equal (that is, sides 1, 3, 5, ... are equal, and sides 2, 4, 6, ... are equal).
- Inscribed circle
- Jung's theorem, an inequality relating the diameter of a point set to the radius of its minimum bounding circle
- Lester's theorem
- Circumscribed sphere
- Triangle center
- Japanese theorem for cyclic quadrilaterals
- Japanese theorem for cyclic polygons
- ^ This can be proven on the grounds that the general equation for a circle with center (a, b) and radius r in the Cartesian coordinate system is
- ^ Dörrie, Heinrich, 100 Great Problems of Elementary Mathematics, Dover, 1965.
- ^ Wolfram page on barycentric coordinates
- ^ a b Nelson, Roger, "Euler's triangle inequality via proof without words," Mathematics Magazine 81(1), February 2008, 58-61.
- ^ Johnson, Roger A., Advanced Euclidean Geometry, Dover, 2007 (orig. 1929), p. 189, #298(d).
- ^ De Villiers, Michael. "Equiangular cyclic and equilateral circumscribed polygons," Mathematical Gazette 95, March 2011, 102-107.
- ^ Megiddo, N. (1983). "Linear-time algorithms for linear programming in R3 and related problems". SIAM Journal on Computing 12 (4): 759–776. doi:10.1137/0212052.
- Kimberling, Clark (1998). "Triangle centers and central triangles". Congressus Numerantium 129: i–xxv, 1–295.
- Derivation of formula for radius of circumcircle of triangle at Mathalino.com
- Weisstein, Eric W., "Circumcircle" from MathWorld.
- Weisstein, Eric W., "Cyclic Polygon" from MathWorld.
- Weisstein, Eric W., "Steiner circumellipse" from MathWorld.
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