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Circumscribed circle

In geometry, a circumscribed circle, also known as a circumcircle, of a polygon is a circle whose circumference passes through each vertex (or angle) of the polygon.^[1] This configuration means the polygon is inscribed in the circle, with all vertices lying on its boundary, while the sides of the polygon lie inside the disk bounded by the circle.^[2] The center of this circle, called the circumcenter, is the point equidistant from all vertices, and the distance from the circumcenter to any vertex is the circumradius. For any triangle, a unique circumcircle exists, with the circumcenter located at the intersection of the perpendicular bisectors of the triangle's sides. This circumcircle encloses the entire triangle, containing all interior points inside or on its boundary.^[3] In the case of a triangle, the circumradius R can be computed using the formula R = \frac{abc}{4K}, where a, b, and c are the side lengths and K is the area of the triangle.^[4] Not all polygons possess a circumcircle; it exists only for cyclic polygons, where all vertices lie on a single circle.^[3] Regular polygons always admit a circumcircle, as their vertices are equally spaced on the circle, with the central angle between adjacent vertices measuring $360^\circ / n for an n-sided polygon.^[2] The circumcenter of a regular polygon coincides with its center of symmetry, formed by the intersection of its angle bisectors.^[2] In Euclidean geometry, these properties underpin theorems in circle geometry, including those related to inscribed angles and cyclic quadrilaterals.^[5]

Definition and Basic Concepts

Definition

A circumscribed circle, also known as a circumcircle, is a circle that passes through all the vertices of a given polygon or a set of points in a plane.^[6] The polygon is described as being inscribed in the circle, meaning its vertices lie on the circumference, with the sides of the polygon lying inside the disk bounded by the circle.^[7] This concept assumes familiarity with basic elements of Euclidean geometry, such as circles—defined as the set of points equidistant from a center—and polygons as closed figures formed by connected line segments.^[8] Points that lie on the same circle are termed concyclic, a property essential to the existence of a circumcircle for the given set.^[9] The center of the circumcircle is called the circumcenter, often denoted as O, and the distance from this center to any vertex is the circumradius, denoted as R.^[8]^[6] In contrast to the incircle, which is tangent to all sides of the polygon and lies entirely within it, the circumcircle passes directly through the vertices and may encompass the polygon's interior.^[10] Excircles, relevant primarily to triangles, are circles tangent to one side and the extensions of the other two sides, positioned outside the figure.^[11] For example, consider a triangle with vertices labeled A, B, and C; its circumcircle is the unique circle passing through these three points, illustrating the general case for any polygon with concyclic vertices.^[8]^[12] Such polygons are known as cyclic, a topic explored further in relation to specific geometric conditions.^[7]

Cyclic Polygons

A cyclic polygon is a polygon whose vertices all lie on the boundary of a single circle, meaning it is inscribed in that circle or, equivalently, the circle is circumscribed around the polygon.^[13] This property ensures the existence of a unique circumcircle for the polygon, provided the vertices are not collinear. Every triangle qualifies as a cyclic polygon, as any three non-collinear points determine a unique circle passing through them.^[13] For polygons with more sides, specific conditions determine cyclicity. In particular, a quadrilateral is cyclic if and only if the sums of its pairs of opposite interior angles each equal 180 degrees.^[14] This criterion, established in ancient geometry, distinguishes cyclic quadrilaterals from non-cyclic ones. An additional property of cyclic quadrilaterals is given by Ptolemy's theorem, which states that the product of the lengths of the diagonals equals the sum of the products of the lengths of the opposite sides.^[15] For polygons with five or more sides, cyclicity requires that all vertices satisfy the circle equation simultaneously, though explicit angle conditions become more complex and are typically verified geometrically or algebraically. A key geometric property of cyclic polygons arises from the inscribed angle theorem: angles subtended by the same arc at the circumference are equal, meaning that if multiple vertices are concyclic, the angles at those vertices intercepting a common arc will be congruent.^[16] This equality holds because such angles share the same central angle measure at the circle's center. Cyclic polygons should not be confused with tangential polygons, which admit an incircle tangent to all sides rather than a circumcircle passing through the vertices; the former emphasizes side tangency, while the latter focuses on vertex inscription.^[17]

Circumcircle of a Triangle

Existence and Uniqueness

A circumscribed circle, or circumcircle, exists for every non-degenerate triangle, meaning any three non-collinear points in the Euclidean plane determine such a circle passing through all three vertices.^[18] This existence follows from the geometric construction of the circle's center as the intersection point of the perpendicular bisectors of the triangle's sides.^[19] Specifically, the perpendicular bisector of one side is the locus of points equidistant from its endpoints, and the intersection of two such bisectors yields a point equidistant from all three vertices, serving as the center.^[20] The uniqueness of this circumcircle arises because the perpendicular bisectors of the sides are concurrent at a single point, ensuring only one such center exists.^[18] To see this, consider the perpendicular bisectors of sides AB and BC intersecting at point O; by SAS congruence in triangles formed with the midpoints, OA = OB and OB = OC, confirming equidistance.^[18] Any other purported center would also lie on these bisectors, implying it coincides with O, thus proving uniqueness by contradiction.^[18] In the degenerate case where the three points are collinear, no finite circumcircle exists, as the perpendicular bisectors would be parallel or coincident, failing to intersect at a single point.^[19] Consequently, the discussion assumes non-degenerate triangles, for which the circumcircle is always well-defined and unique. All triangles are inherently cyclic polygons, possessing this circumcircle by virtue of having exactly three sides.^[18]

Circumcenter and Circumradius

The circumcenter of a triangle, denoted as O, is the point where the perpendicular bisectors of the three sides intersect. This point serves as the center of the circumscribed circle, which passes through all three vertices of the triangle, and it is equidistant from each vertex, with the common distance being the circumradius R. Thus, OA = OB = OC = R, where A, B, and C are the vertices.^[4]^[21] The circumradius R is defined as the distance from the circumcenter O to any of the triangle's vertices. A fundamental relation connecting R to the triangle's sides and angles is given by the extended law of sines, which states that R = \frac{a}{2 \sin A} = \frac{b}{2 \sin B} = \frac{c}{2 \sin C}, where a, b, and c are the lengths of the sides opposite angles A, B, and C, respectively.^[22] In specific triangle types, the circumcenter exhibits notable properties. For an equilateral triangle, the circumcenter coincides with the centroid, as well as the orthocenter, due to the symmetry of the figure. In a right triangle, Thales' theorem implies that the hypotenuse serves as the diameter of the circumcircle, positioning the circumcenter at the midpoint of the hypotenuse and setting R equal to half the hypotenuse length.^[23]^[24] The circumcenter plays a key role in the Euler line of the triangle, which is the straight line passing through the circumcenter O, the centroid G, and the orthocenter H. On this line, the centroid divides the segment from the orthocenter to the circumcenter in the ratio 2:1, with G closer to O.^[25]

Construction Methods

Straightedge and Compass Construction

The classical method for constructing the circumcircle of a triangle relies on finding the circumcenter, which is the intersection point of the perpendicular bisectors of the triangle's sides, using only a straightedge and a compass.^[26] A straightedge allows drawing straight lines between points or extending lines indefinitely, while the compass enables drawing circles of specified radii and transferring distances.^[27] To perform the construction for a given triangle ABC, begin by constructing the perpendicular bisectors of at least two sides, say AB and AC. To construct the perpendicular bisector of AB, open the compass to a radius greater than half the length of AB. Place the compass point at A and draw arcs above and below the line AB. Without changing the radius, place the compass point at B and draw arcs intersecting the previous arcs at two points. Draw the line through these two intersection points with the straightedge; this line is the perpendicular bisector of AB, intersecting AB at its midpoint D. Similarly, construct the perpendicular bisector of AC, intersecting AC at its midpoint E. The intersection point O of these two perpendicular bisectors serves as the circumcenter. Finally, set the compass to the distance from O to A (or any vertex), place the point at O, and draw the circle, which will pass through A, B, and C.^[26] Verification of the construction involves drawing the perpendicular bisector of the third side BC; it will intersect the previous bisectors at the same point O, confirming that O is equidistant from all three vertices and thus the unique circumcenter.^[26] This property holds because any point on a perpendicular bisector is equidistant from the endpoints of the segment it bisects, ensuring OA = OB = OC. This method originates from ancient Greek geometry and is detailed in Euclid's Elements, specifically Book IV, Proposition 5, which proves that the circle centered at the intersection of two such bisectors circumscribes the triangle.^[26] Alternative constructions exist for special cases, such as right triangles, but the perpendicular bisector approach applies universally to any triangle.

Alternative Constructions

One alternative method for constructing the circumcircle of a triangle exploits angular properties derived from circle theorems. From vertex A of triangle ABC, construct a line departing from side AB (or AC) at an angle equal to 90° minus the measure of angle C (or angle B, respectively). Repeat this process from another vertex, such as B, drawing a line from side BA at an angle of 90° minus angle A. The intersection point of these lines locates the circumcenter O, from which the circumcircle can be drawn with radius equal to the distance from O to any vertex. This approach relies on the geometric relationship that positions O equidistant from the vertices, and for obtuse triangles where the opposite angle exceeds 90°, the line is extended outward in the negative angular direction.^[28] For right-angled triangles, a particularly efficient construction follows directly from Thales' theorem, which states that if a triangle is inscribed in a circle where one side of the triangle is a diameter of the circle, then the opposite angle is a right angle; conversely, for a right triangle with the right angle at C, the hypotenuse AB serves as the diameter of the circumcircle. Thus, the circumcenter O is simply the midpoint of the hypotenuse AB, which can be constructed by finding the perpendicular bisector's midpoint using straightedge and compass. The circumcircle is then drawn with center O and radius OA (or OB). This method avoids constructing full perpendicular bisectors of all sides, as the theorem guarantees the configuration.^[29] Another illustrative approach involves reflections and projections, such as reflecting the orthocenter over the triangle's sides to obtain points that lie on the circumcircle; these reflected points can aid in verifying or sketching the circle in advanced geometric diagrams, though it is not a primary construction tool for locating O.^[30] These alternative methods offer advantages in speed and simplicity for specific triangle types, such as right or obtuse configurations, by bypassing the need to draw complete perpendicular bisectors from the previous general construction, while still yielding the precise circumcircle.

Geometric Properties

Location of the Circumcenter

The location of the circumcenter O of a triangle varies depending on whether the triangle is acute, right, or obtuse. In an acute triangle, all angles are less than $90^\circ, and O lies inside the triangle.^[25] In a right triangle, O is located at the midpoint of the hypotenuse, which serves as the diameter of the circumcircle.^[31] In an obtuse triangle, one angle exceeds $90^\circ, and O lies outside the triangle on the side opposite the obtuse angle.^[25] By definition, the distance from O to each vertex equals the circumradius R.^[25] The circumcenter O lies on the Euler line, which connects it to the orthocenter, centroid, and other triangle centers.^[25] Positionally, O relates to the nine-point circle, whose center is the midpoint between O and the orthocenter.^[25]

Angles and Inscribed Angle Theorems

The inscribed angle theorem states that the measure of an inscribed angle in a circle is half the measure of the central angle that subtends the same arc.^[32] In the context of a triangle inscribed in its circumcircle, this means that an angle at a vertex, such as \angle ABC, subtends the arc AC and equals half the central angle at the circumcenter subtending the same arc AC.^[32] This relationship holds because the inscribed angle intercepts the arc between the endpoints of its sides, while the central angle directly measures that arc's extent.^[32] A special case of the inscribed angle theorem occurs when the arc is a semicircle, resulting in a right angle at the circumference.^[33] For instance, if AB is the diameter of the circumcircle and C lies on the circumference, then \angle ACB = 90^\circ, as the central angle subtending the semicircle arc AB is $180^\circ.^[33] This theorem, often attributed to Thales, provides a foundational example of how the circumcircle constrains triangular angles.^[33] The alternate segment theorem complements these ideas by relating angles formed by a tangent to the circle and a chord to inscribed angles in the alternate segment.^[34] Specifically, if a tangent at point B meets chord BD, the angle between the tangent and chord equals the inscribed angle subtended by BD in the segment not containing the tangent point.^[34] For a triangle ABC with circumcircle, drawing a tangent at B to chord BA shows that the tangent-chord angle equals \angle BCA in the alternate segment.^[34] These angle theorems underpin key trigonometric relations in triangles, such as the extended law of sines.^[35] The inscribed angle theorem implies that vertex \angle A subtends arc BC of measure $2\angle A, so the central angle is $2A; this arc measure directly leads to the side opposite \angle A relating to the circumradius via the sine of the central angle halved.^[35] Thus, the theorems explain the proportionality a / \sin A = 2R, where R is the circumradius, by linking angular measures across the circle.^[35]

Relations to Other Triangle Centers

In any triangle, the circumcenter O, the orthocenter H, and the centroid G are collinear on the Euler line, with the centroid dividing the segment from the orthocenter to the circumcenter in the ratio HG = 2GO.^[36] The squared distance between the circumcenter and orthocenter is given by OH^2 = 9R^2 - (a^2 + b^2 + c^2), where R is the circumradius and a, b, c are the side lengths.^[37] Similarly, the squared distance between the circumcenter and incenter I is OI^2 = R(R - 2r), where r is the inradius.^[38] The nine-point circle, which passes through the midpoints of the sides, the feet of the altitudes, and the midpoints of the segments from the orthocenter to the vertices, has its center at the midpoint of the segment joining the circumcenter and orthocenter, and its radius is R/2.^[39] In an equilateral triangle, the circumcenter, orthocenter, centroid, and incenter all coincide at the same point.^[38] A key relation involves the orthocenter: the reflections of the orthocenter over the three sides of the triangle lie on the circumcircle.^[30]

Formulas and Equations

Circumradius Formulas

The circumradius R of a triangle can be computed using the extended law of sines, which relates the side lengths to the opposite angles and the circumradius. For a triangle with sides a, b, c opposite angles A, B, C respectively, the formula is

R = \frac{a}{2 \sin A} = \frac{b}{2 \sin B} = \frac{c}{2 \sin C}.

This expression arises from the inscribed angle theorem, which states that an inscribed angle is half the central angle subtending the same arc. The central angle subtending arc BC (opposite vertex A) is $2A, and the chord length a (side BC) satisfies a = 2R \sin A, yielding R = a / (2 \sin A ).^[40]^[41] Another fundamental formula expresses R in terms of the side lengths and the area K of the triangle:

R = \frac{abc}{4K}.

This follows from the extended law of sines: a = 2R \sin A, b = 2R \sin B, c = 2R \sin C, so abc = 8 R^3 \sin A \sin B \sin C. The area K = \frac{1}{2} bc \sin A = 2 R^2 \sin B \sin C \sin A, thus abc = 4 R K, yielding R = abc / (4K). Here, K can be computed using Heron's formula, K = \sqrt{s(s-a)(s-b)(s-c)}, where s = (a + b + c)/2 is the semiperimeter.^[42]^[43] Equivalently, without explicit area computation,

R = \frac{abc}{\sqrt{(a+b+c)(b+c-a)(c+a-b)(a+b-c)}}.

This formula derives directly from substituting Heron's expression into R = abc / (4K).^[42]

Circumcenter Coordinates

The circumcenter of a triangle can be located using coordinate geometry by solving for the intersection of the perpendicular bisectors of the sides, assuming the vertices are given as A(a_x, a_y), B(b_x, b_y), and C(c_x, c_y) in the Cartesian plane.^[38] The x-coordinate of the circumcenter O is given by

O_x = \frac{(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)}{D},

and the y-coordinate by

O_y = \frac{(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)}{D},

where the denominator is

D = 2 \left[ a_x (b_y - c_y) + b_x (c_y - a_y) + c_x (a_y - b_y) \right].

This formula arises from setting up the equations of two perpendicular bisectors and solving the resulting linear system, ensuring O is equidistant from A, B, and C.^[44] An equivalent determinant form expresses the coordinates as ratios of 3×3 determinants:

O_x = \frac{ \det \begin{vmatrix} a_x^2 + a_y^2 & a_y & 1 \\ b_x^2 + b_y^2 & b_y & 1 \\ c_x^2 + c_y^2 & c_y & 1 \end{vmatrix} }{ 2 \det \begin{vmatrix} a_x & a_y & 1 \\ b_x & b_y & 1 \\ c_x & c_y & 1 \end{vmatrix} },

O_y = \frac{ \det \begin{vmatrix} a_x & a_x^2 + a_y^2 & 1 \\ b_x & b_x^2 + b_y^2 & 1 \\ c_x & c_x^2 + c_y^2 & 1 \end{vmatrix} }{ 2 \det \begin{vmatrix} a_x & a_y & 1 \\ b_x & b_y & 1 \\ c_x & c_y & 1 \end{vmatrix} }.

The denominator in both cases is twice the signed area of the triangle.^[45] In trilinear coordinates with respect to triangle ABC, the circumcenter has coordinates \cos A : \cos B : \cos C, where A, B, C are the angles at the respective vertices.^[38] This reflects the circumcenter's position as the point where the distances to the sides are proportional to the cosines of the opposite angles. An equivalent form is a(b^2 + c^2 - a^2) : b(c^2 + a^2 - b^2) : c(a^2 + b^2 - c^2), where a, b, c are the side lengths opposite A, B, C, since b^2 + c^2 - a^2 = 2bc \cos A.^[45] The barycentric coordinates of the circumcenter are \sin 2A : \sin 2B : \sin 2C.^[46] These can be expressed without angles as a^2 (b^2 + c^2 - a^2) : b^2 (c^2 + a^2 - b^2) : c^2 (a^2 + b^2 - c^2), allowing computation from side lengths alone; the actual position O is then the weighted average O = (\alpha A + \beta B + \gamma C) / (\alpha + \beta + \gamma), where \alpha = a^2 (b^2 + c^2 - a^2), and similarly for \beta, \gamma by cyclic permutation.^[45] For a triangle with position vectors \mathbf{A}, \mathbf{B}, \mathbf{C}, where a = |\mathbf{B} - \mathbf{C}|, b = |\mathbf{C} - \mathbf{A}|, c = |\mathbf{A} - \mathbf{B}|, the circumcenter \mathbf{O} is given by

\mathbf{O} = \frac{ a^2 (\mathbf{B} - \mathbf{C}) + b^2 (\mathbf{C} - \mathbf{A}) + c^2 (\mathbf{A} - \mathbf{B}) }{ 2 ( a^2 (\mathbf{B} - \mathbf{C}) + b^2 (\mathbf{C} - \mathbf{A}) + c^2 (\mathbf{A} - \mathbf{B}) ) \cdot \mathbf{n} },

adjusted for the plane normal \mathbf{n}, but more precisely using the barycentric form above projected in 3D space.

Equation of the Circumcircle

The general equation of a circle in the Cartesian plane is given by

x^2 + y^2 + Dx + Ey + F = 0,

where D, E, and F are constants to be determined. For the circumcircle of a triangle with vertices A(x_1, y_1), B(x_2, y_2), and C(x_3, y_3), substitute these points into the equation to form a system of three linear equations in D, E, and F:

\begin{align*} x_1^2 + y_1^2 + D x_1 + E y_1 + F &= 0, \\ x_2^2 + y_2^2 + D x_2 + E y_2 + F &= 0, \\ x_3^2 + y_3^2 + D x_3 + E y_3 + F &= 0. \end{align*}

Solving this system yields the specific equation of the circumcircle.^[47] An alternative representation uses a determinant, which directly gives the equation without solving for the coefficients explicitly. For vertices (x_i, y_i) with i = 1, 2, 3, the circumcircle equation is

\begin{vmatrix} x^2 + y^2 & x & y & 1 \\ x_1^2 + y_1^2 & x_1 & y_1 & 1 \\ x_2^2 + y_2^2 & x_2 & y_2 & 1 \\ x_3^2 + y_3^2 & x_3 & y_3 & 1 \end{vmatrix} = 0.

Expanding this determinant produces the Cartesian equation equivalent to the general form above.^[8] The parametric equation of the circumcircle, centered at the circumcenter O(h, k) with radius R, is

(x, y) = (h + R \cos \theta, k + R \sin \theta),

where the parameter \theta is chosen such that the points corresponding to the vertices A, B, and C lie on the circle, typically by aligning \theta with the angular positions of the vertices relative to O.^[8] In barycentric coordinates (x : y : z) with respect to triangle ABC (where a = BC, b = CA, c = AB), the homogeneous equation of the circumcircle is

a^2 y z + b^2 z x + c^2 x y = 0.

This quadratic form arises from the general equation of a conic in barycentric coordinates, specialized to pass through the vertices A(1:0:0), B(0:1:0), and C(0:0:1). For normalized barycentric coordinates where x + y + z = 1, the equation remains homogeneous and defines the circle in the plane.^[46]^[48] The corresponding form in trilinear coordinates, which are scaled by the side lengths relative to barycentric (i.e., trilinear (\alpha : \beta : \gamma) = (a x : b y : c z)), adjusts to

a y z + b z x + c x y = 0,

reflecting the linear scaling in the coordinate system while preserving the conic nature of the circumcircle.^[49] In higher dimensions, the equation of the circumsphere of a tetrahedron with vertices (x_i, y_i, z_i) for i = 1, 2, 3, 4 generalizes the determinant form to

\begin{vmatrix} x^2 + y^2 + z^2 & x & y & z & 1 \\ x_1^2 + y_1^2 + z_1^2 & x_1 & y_1 & z_1 & 1 \\ x_2^2 + y_2^2 + z_2^2 & x_2 & y_2 & z_2 & 1 \\ x_3^2 + y_3^2 + z_3^2 & x_3 & y_3 & z_3 & 1 \\ x_4^2 + y_4^2 + z_4^2 & x_4 & y_4 & z_4 & 1 \end{vmatrix} = 0.

This defines the unique sphere passing through the four non-coplanar points, analogous to the planar case.^[50]

Generalization to Other Polygons

Cyclic Quadrilaterals

A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a single circle, allowing a circumscribed circle to pass through each vertex. A fundamental property is that the sum of each pair of opposite interior angles equals 180 degrees, which serves as both a necessary and sufficient condition for a quadrilateral to be cyclic.^[51] The area K of a cyclic quadrilateral with side lengths a, b, c, and d is given by Brahmagupta's formula:

K = \sqrt{(s - a)(s - b)(s - c)(s - d)},

where s = \frac{a + b + c + d}{2} is the semiperimeter; this formula yields the maximum possible area for given side lengths among all quadrilaterals.^[52] The circumradius R of a cyclic quadrilateral can be computed using the formula

R = \frac{\sqrt{(ab + cd)(ac + bd)(ad + bc)}}{4K},

where K is the area from Brahmagupta's formula; this expression derives from relating the extended law of sines to the quadrilateral's geometry.^[51] Ptolemy's theorem provides a key relation for cyclic quadrilaterals: the product of the lengths of the two diagonals equals the sum of the products of the lengths of the opposite sides, i.e., ac + bd = pq, where p and q are the diagonals; this equality holds specifically for cyclic quadrilaterals and characterizes them among all quadrilaterals.^[15] Certain quadrilaterals are inherently cyclic, such as any rectangle, where all angles are 90 degrees and thus opposite angles sum to 180 degrees, and any isosceles trapezoid, which has a pair of parallel sides and equal non-parallel sides ensuring the base angles are equal and supplementary to the opposite angles. The circumcenter of a cyclic quadrilateral lies at the intersection of the perpendicular bisectors of its sides, consistent with the general property for cyclic polygons.^[51]

Higher-Order Cyclic Polygons

A cyclic n-gon, for n > 4, is a polygon whose vertices lie on a common circumcircle, generalizing the concept from triangles and quadrilaterals. Unlike triangles, which always admit a circumcircle, or quadrilaterals where specific conditions like the sum of opposite angles equaling 180 degrees suffice, higher-order polygons do not necessarily possess one; the vertices must all satisfy the general equation of a circle, (x - h)^2 + (y - k)^2 = r^2 for some center (h, k) and radius r. For n ≥ 5, a set of n points in the plane forms a cyclic polygon if and only if every subset of four points is concyclic, providing a recursive verification method based on quadrilateral conditions.^[53]^[13] Regular n-gons are always cyclic, with vertices equally spaced on the circumcircle by construction. The circumradius R of a regular n-gon with side length s is R = \frac{s}{2 \sin(\pi / n)}. This formula derives from the central angle of 2\pi / n between adjacent vertices, forming an isosceles triangle with two radii and base s.^[42]^[2] For pentagons, cyclic pentagons lack a simple angle sum condition analogous to quadrilaterals; instead, properties are verified using trigonometric identities relating side lengths and angles or by embedding in the complex plane, where concyclicity holds if the points' arguments satisfy certain harmonic properties. For example, a cyclic pentagon requires its vertices to meet the concyclic criterion, often imposing constraints that reduce the degrees of freedom compared to general pentagons. Not all pentagons are cyclic; for instance, a non-convex pentagon whose vertices do not lie on a single circle cannot have a circumcircle, highlighting the restrictive nature of the condition for higher n.^[54]^[55]

Applications and Extensions

In Euclidean Geometry

In Euclidean geometry, the circumscribed circle, or circumcircle, of a triangle is instrumental in proofs of similarity and concurrency. The AA similarity criterion for triangles can be established through properties of inscribed angles, where angles subtending the same arc on the circumcircle are equal, enabling the identification of corresponding angles in distinct triangles sharing such arc relationships.^[56] Additionally, the perpendicular bisectors of a triangle's sides are concurrent at the circumcenter, the point equidistant from all vertices and thus the center of the circumcircle, forming a foundational concurrency theorem that defines the circle's existence and position.^[57] The circumcircle also underpins key trigonometric relations, particularly the extended law of sines. For a triangle with sides a, b, c opposite angles A, B, C and circumradius R,

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R.

This derives from the isosceles triangle formed by two radii to the endpoints of side a and the center, where the central angle is $2A, yielding a = 2R \sin A via the law of sines applied to that triangle.^[58] Constructions involving the circumcircle enable the division of a circle into equal arcs, which is essential for inscribing regular polygons; vertices are placed successively at arc endpoints using compass settings equal to the chord length corresponding to the central angle $360^\circ / n for an n-gon.^[4] In triangle classification, Thales' theorem identifies right triangles: if a triangle is inscribed in a circle with one side as the diameter, the opposite angle is a right angle, and conversely, the hypotenuse of a right triangle serves as the diameter of its circumcircle.^[59]

In Non-Euclidean and Higher-Dimensional Geometry

In spherical geometry, the circumcircle of a spherical triangle is defined as the small circle (or great circle, if applicable) on the sphere that passes through its three vertices. This circle lies on a plane intersecting the sphere, and its radius is determined by the distance from the sphere's center to that plane. Unlike in Euclidean geometry, where any three non-collinear points determine a unique circle, in spherical geometry the circumcircle is always unique for any spherical triangle.^[60] In hyperbolic geometry, not every triangle admits a circumcircle; existence requires that the longest side c satisfies \sinh(c/2) < \sinh(a/2) + \sinh(b/2), where a \leq b \leq c are the side lengths. When this condition holds, the vertices lie on a hyperbolic circle, a curve of constant positive curvature. If equality obtains, the vertices lie on a horocycle, a curve of zero curvature tangent to the boundary at infinity; if strict inequality in the opposite direction, they lie on a hypercycle, a curve equidistant from a geodesic. This classification arises because hyperbolic circles are limits of Euclidean circles in models like the Poincaré disk, and horocycles serve as the boundary case where the center escapes to infinity. Any three non-collinear points in the hyperbolic plane lie on exactly one such circline (circle, horocycle, or hypercycle).^[61]^[62] The concept of a circumscribed circle generalizes to higher dimensions as the circumhypersphere, the unique hypersphere passing through all vertices of a simplex, assuming the points are in general position. For a tetrahedron in three dimensions—the 3-simplex—the circumsphere's center and radius can be computed via a determinant formula involving the vertices' coordinates (x_i, y_i, z_i) for i=1 to $4:

\begin{vmatrix} x^2 + y^2 + z^2 & x & y & z & 1 \\ x_1^2 + y_1^2 + z_1^2 & x_1 & y_1 & z_1 & 1 \\ x_2^2 + y_2^2 + z_2^2 & x_2 & y_2 & z_2 & 1 \\ x_3^2 + y_3^2 + z_3^2 & x_3 & y_3 & z_3 & 1 \\ x_4^2 + y_4^2 + z_4^2 & x_4 & y_4 & z_4 & 1 \end{vmatrix} = 0,

which expands to a(x^2 + y^2 + z^2) - (D_x x + D_y y + D_z z) + D = 0, where a, D_x, D_y, D_z, D are 4×4 determinants of the coordinates and squared distances. The center is at (D_x/(2a), D_y/(2a), D_z/(2a)), and the radius is \sqrt{(D_x^2 + D_y^2 + D_z^2 - 4aD)/(4a^2)}. This determinant approach extends to n-dimensions for the circumhypersphere of an n-simplex, solving a system of quadratic equations via the Cayley-Menger determinant, which encodes all pairwise distances and yields the center as barycentric coordinates weighted by subdeterminants. Uniqueness holds if the simplex is non-degenerate, but in non-Euclidean higher-dimensional spaces, such as hyperbolic 3-manifolds, existence may fail analogously if points violate curvature-constrained conditions.^[50]^[63] Applications of circumhyperspheres appear in computer graphics for 3D modeling, where the smallest enclosing sphere (often the circumsphere for simplices) serves as a bounding volume to accelerate collision detection and rendering in polygonal meshes; for instance, in Delaunay triangulation algorithms, circumspheres determine tetrahedral validity by checking empty sphere criteria. In GPS systems, spherical Earth approximations model satellite ranging as intersections of spheres, yielding positions on the Earth's surface where great circles act as geodesics, and small circles approximate circumcircles for local triangular networks in differential positioning, though oblate ellipsoid corrections are applied for precision. Limitations in non-Euclidean settings include non-uniqueness or non-existence: on spheres, circumcircles may not be unique for hemispherical triangles due to dual intersections, while in hyperbolic spaces, the failure condition \sinh(c/2) \geq \sinh(a/2) + \sinh(b/2) precludes a proper circumcircle, resorting to horocycles that lack a finite center.^[64]^[65]

History

Ancient Contributions

The earliest known contributions to the concept of the circumscribed circle emerged in ancient Greek mathematics around the 6th century BCE. Thales of Miletus (c. 624–546 BCE) is attributed with discovering the theorem that a triangle inscribed in a circle, with one side as the diameter, forms a right angle at the third vertex, establishing the circumcircle's role in right triangles.^[66] This insight, preserved through later accounts, marked an initial geometric application of circular properties to triangular configurations.^[67] Euclid of Alexandria (c. 300 BCE) systematized these ideas in his Elements, particularly in Books III and IV, where he explored circles, inscribed angles, and polygons inscribed within or circumscribed about circles.^[68] These propositions defined key relationships, such as angles subtended by the same arc and the inscription of regular polygons in circles, providing a rigorous framework for circumcircles without explicit formulas for radius or center.^[69] Hippocrates of Chios (c. 470–410 BCE) extended these principles in his work on the quadrature of lunes, crescent-shaped regions bounded by two circular arcs, where he demonstrated that certain lunes could be squared exactly using properties of circumcircles around squares and semicircles.^[70] His methods, the earliest surviving geometric proofs involving such figures, highlighted the circumcircle's utility in area comparisons between circular and rectilinear shapes.^[71] In Indian mathematics, Brahmagupta (c. 598–668 CE) advanced the study of cyclic quadrilaterals—figures inscribed in a circle—in his Brahma-sphuta-siddhanta (628 CE), providing a formula for their area and applying these properties in astronomical computations for planetary positions and eclipse predictions.^[52] This work built on earlier geometric traditions and influenced subsequent Indian astronomical texts.^[72] Claudius Ptolemy (c. 100–170 CE) further advanced cyclic properties in his Almagest (c. 150 CE), employing them in spherical trigonometry for astronomical calculations, including the theorem relating sides and diagonals of cyclic quadrilaterals to derive chord lengths and angular measures.^[73] These developments synthesized Greek geometric traditions with practical celestial modeling, sustaining their influence through the medieval period.^[74]

Modern Developments

In the 19th century, Leonhard Euler's foundational 1765 work on triangle centers, which established the collinearity of the orthocenter, centroid, and circumcenter along the Euler line, received more formalized analytical treatment amid advances in coordinate geometry.^[75] This development highlighted the circumcenter's central role in unifying triangle properties through algebraic methods.^[76] Concurrently, Gaspard Monge's contributions in the 1790s to the geometry of tetrahedrons introduced concepts related to the Monge point, which interacts with the circumsphere as the reflection of the centroid over the orthocenter in higher-dimensional analogs.^[77] Arthur Cayley's introduction of the Cayley-Menger determinant in the 1840s provided a determinant-based formula using pairwise distances to compute the circumradius of simplices in higher dimensions, enabling precise volume and sphere calculations without coordinates.^[78]^[79] The 20th century saw the circumcircle integrated into computational geometry, particularly through Boris Delaunay's 1934 definition of the Delaunay triangulation, where triangles are formed such that no point lies inside the circumcircle of any triangle, ensuring optimal mesh quality.^[80] Algorithms like the Bowyer-Watson method, developed in 1981, efficiently compute these triangulations by incrementally inserting points while maintaining the empty circumcircle property, with applications in finite element methods for simulating physical phenomena such as stress analysis and fluid dynamics.^[81] These techniques revolutionized numerical simulations by producing meshes that minimize angular distortion and enhance convergence rates in partial differential equation solvers.^[82] In contemporary applications, circumcircles facilitate robotics path planning by modeling robot footprints as circumscribed circles for collision avoidance; for instance, in edge coverage tasks like robotic mowing, obstacles are dilated by the robot's circumcircle to generate safe trajectories.^[83] Similarly, dynamic geometry software such as GeoGebra enables interactive constructions of circumcircles, allowing users to manipulate triangle vertices in real-time to explore properties like the circumcenter's locus and radius variations. This tool supports educational and research visualizations, integrating circumcircle computations with algebraic solvers for precise dynamic feedback.^[84]

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