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Concyclic points

In geometry, concyclic points (or cocyclic points) are a set of two or more points that lie on the circumference of a single circle in the Euclidean plane.^[1] This property is fundamental to circle geometry, as it allows for the analysis of configurations where points share a common circumcircle.^[2] Any two distinct points are trivially concyclic, since infinitely many circles can pass through them, but the concept becomes nontrivial for three or more points.^[1] For three non-collinear points, they always determine a unique circle, making them concyclic by definition; this circumcircle is the unique circle passing through all three.^[1] In the case of four points, determining concyclicity requires specific conditions, such as the points forming a quadrilateral whose opposite interior angles sum to 180 degrees—a hallmark property of cyclic quadrilaterals.^[3] Alternatively, Ptolemy's theorem provides a metric criterion: for a quadrilateral with sides a, b, c, d and diagonals p, q, the points are concyclic if and only if ac + bd = pq.^[1] These conditions enable verification of concyclicity without directly constructing the circle. Concyclic points play a crucial role in various geometric theorems and applications, including the study of polygons with maximum area for given side lengths (where making consecutive points concyclic increases the area) and in trigonometric identities involving inscribed angles.^[1]^[4] They also appear in advanced topics, such as lattice point configurations, where the absence of four concyclic points relates to unsolved problems in number theory.^[1]

Fundamentals

Definition and Characterization

In Euclidean geometry, a set of points in the plane are said to be concyclic if they all lie on the same circle. Specifically, four or more distinct points P_1, P_2, P_3, P_4, \dots are concyclic if there exists a unique circle passing through all of them. For three points, they are always concyclic provided they are not collinear, as any three non-collinear points determine a unique circle.^[1]^[5] A key characterization of concyclic points is that the perpendicular bisectors of the line segments joining pairs of these points intersect at a single common point, known as the circumcenter. This circumcenter is equidistant from all the points, serving as the center of the circle on which they lie. The equation of such a circle, with center (h, k) and radius r, is given by

(x - h)^2 + (y - k)^2 = r^2,

where the coordinates of each concyclic point satisfy this equation, and the distance from the circumcenter to any point equals r.^[6]^[7] An illustrative example of concyclic points is the vertices of an equilateral triangle, which lie on a circle whose center is the centroid (also the circumcenter) of the triangle. In this configuration, the perpendicular bisectors of the sides coincide with the altitudes and medians, all intersecting at the circumcenter.

Construction via Perpendicular Bisectors

One effective geometric construction to locate the circumcircle of three non-collinear points A, B, and C—and thereby determine if they are concyclic—involves the perpendicular bisectors of the segments connecting them. The perpendicular bisector of a line segment is the line that passes through its midpoint and is perpendicular to it, consisting of all points equidistant from the segment's endpoints.^[8] To construct the circumcenter O, draw the perpendicular bisector of AB and the perpendicular bisector of BC; their intersection point O is equidistant from A, B, and C, serving as the center of the unique circle passing through all three points.^[8] Verify concyclicity by confirming that the distances OA, OB, and OC are equal, which defines the radius R of the circumcircle.^[9] For more than three points, such as four points A, B, C, and D, extend the construction by first finding the circumcenter O using any three non-collinear points (e.g., A, B, C), then check if the fourth point D is equidistant from O at distance R. Equivalently, compute the perpendicular bisectors of multiple segments (e.g., AB, BC, and CD) and verify if they all intersect at a single point; concurrence confirms that all points lie on the same circle.^[9] This method leverages the property that the circumcenter is the intersection of any two perpendicular bisectors, with additional bisectors providing verification.^[8] In coordinate geometry, the equation of the perpendicular bisector of two points (x_1, y_1) and (x_2, y_2) can be derived as follows. First, compute the midpoint M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). The slope of the segment is m = \frac{y_2 - y_1}{x_2 - x_1} (assuming x_1 \neq x_2), so the perpendicular slope is m_\perp = -\frac{x_2 - x_1}{y_2 - y_1} (or m_\perp = 0 if vertical). The equation is then y - y_M = m_\perp (x - x_M), which simplifies to the line passing through M with slope m_\perp.^[10] For vertical or horizontal segments, use the appropriate form (e.g., x = x_M for vertical). To find O, solve the system of equations from two such bisectors. Consider an example with a scalene triangle having vertices A(0,0), B(4,0), and C(1,3). The perpendicular bisector of AB (midpoint (2,0), horizontal segment so vertical line) is x = 2. The perpendicular bisector of BC has midpoint \left( \frac{5}{2}, \frac{3}{2} \right) and segment slope \frac{3-0}{1-4} = -1, so perpendicular slope is 1; its equation is y - \frac{3}{2} = 1 \left( x - \frac{5}{2} \right), or y = x - 1. Intersecting with x=2 gives O(2, 1). The radius R = distance OA = \sqrt{(2-0)^2 + (1-0)^2} = \sqrt{5}, and similarly OB = OC = \sqrt{5}, confirming the circumcircle centered at (2, 1) with radius \sqrt{5}. To check a fourth point D(3,4), compute OD = \sqrt{(3-2)^2 + (4-1)^2} = \sqrt{10} \neq \sqrt{5}, so D is not concyclic with A, B, C.^[9]

Applications in Triangles

Circumcircle and Triangle Vertices

In geometry, the vertices of any non-collinear triangle are always concyclic, lying on a unique circle known as the circumcircle, which passes through all three vertices. This circumcircle is determined by the triangle's vertices and serves as the foundational example of concyclicity in triangular configurations. The center of this circle, called the circumcenter, is the intersection point of the perpendicular bisectors of the triangle's sides.^[11] The radius of the circumcircle, denoted as R, can be calculated using the formula R = \frac{a}{2 \sin A}, where a is the length of a side opposite the angle A in the triangle. This formula derives from the extended law of sines and provides a direct relationship between the triangle's angles, sides, and the circumradius. Unlike the incircle, which is tangent to the three sides of the triangle and touches them at distinct points, the circumcircle passes directly through the vertices without regard to the sides themselves.^[11] A notable example occurs in right-angled triangles, where Thales' theorem states that if the hypotenuse serves as the diameter of the circumcircle, the right angle at the third vertex is subtended by this diameter. In such cases, the circumradius R equals half the hypotenuse length, illustrating a direct geometric consequence of the theorem. This configuration highlights the circumcircle's role in right triangles and underscores the inherent concyclicity of their vertices.^[11]

Additional Concyclic Points in Triangles

In triangle geometry, beyond the vertices lying on the circumcircle, several additional notable points are concyclic on the nine-point circle, a significant circle associated with any given triangle. This circle, discovered by Leonhard Euler in 1765, initially passes through the midpoints of the three sides and the feet of the three altitudes from the vertices to the opposite sides. In 1822, Karl Wilhelm Feuerbach extended the result by showing that it also passes through the three Euler points, which are the midpoints of the segments joining each vertex to the orthocenter (the intersection point of the altitudes).^[12] These nine points—three midpoints of the sides, three feet of the altitudes, and three Euler points—lie on the nine-point circle for any triangle, providing a key example of concyclicity independent of the triangle's type.^[12] The nine-point circle has a radius equal to half the circumradius R of the reference triangle, and its center, known as the nine-point center, is the midpoint of the segment joining the orthocenter and the circumcenter.^[12] A fundamental theorem states that the nine-point circle serves as the circumcircle of the medial triangle, which is formed by connecting the midpoints of the original triangle's sides; it is also the circumcircle of the orthic triangle, formed by the feet of the altitudes.^[13] This property highlights the circle's role in unifying various midpoint and altitude-related configurations. Additionally, certain reflection points, such as the reflections of the orthocenter over the sides, appear in related concyclic sets but typically lie on the circumcircle rather than the nine-point circle.^[14] For an acute triangle, all nine points on the nine-point circle lie inside the triangle, illustrating a compact concyclic arrangement that aids in visualizing these geometric relations.^[12] This configuration contrasts with obtuse triangles, where some points may fall outside, yet the concyclicity persists. The nine-point circle thus exemplifies how non-vertex points in a triangle can share a common circle, extending classical properties of triangle geometry.

Cyclic Quadrilaterals

Defining Properties

A cyclic quadrilateral possesses several defining geometric properties that characterize its vertices as concyclic, meaning they lie on a common circle. One fundamental property is that the sum of each pair of opposite interior angles equals 180 degrees. For a quadrilateral ABCD inscribed in a circle, this means ∠DAB + ∠BCD = 180° and ∠ABC + ∠CDA = 180°.^[15] Another key property involves the exterior angles: the measure of an exterior angle formed by extending one side of the quadrilateral is equal to the measure of the interior angle at the opposite vertex.^[16] This holds because the exterior angle and the opposite interior angle subtend the same arc, ensuring their equality in the cyclic configuration.^[16] Ptolemy's theorem provides an additional defining relation among the side lengths and diagonals, stating that the product of the lengths of the diagonals equals the sum of the products of the lengths of the opposite sides pairs.^[17] This property uniquely identifies cyclic quadrilaterals through their side and diagonal measurements. Common examples of cyclic quadrilaterals include rectangles, where all angles are 90 degrees and thus opposite angles sum to 180 degrees, and isosceles trapezoids, which have pairs of equal base angles that also satisfy the supplementary opposite angle condition.^[15]

Key Theorems and Formulas

One of the most fundamental theorems for cyclic quadrilaterals is Ptolemy's theorem, which relates the lengths of the sides and diagonals. For a cyclic quadrilateral ABCD with sides AB = a, BC = b, CD = c, DA = d, and diagonals AC = p, BD = q, the theorem states that the product of the diagonals equals the sum of the products of the opposite sides:

p \cdot q = ac + bd.

This result, originally described by the Alexandrian mathematician Claudius Ptolemy in the 2nd century CE in his work Almagest, provides a direct way to compute one diagonal if the other three sides and the remaining diagonal are known. A key application of Ptolemy's theorem is in computing diagonal lengths. For example, consider a cyclic quadrilateral with sides a = 3, b = 4, c = 5, d = 6, and one diagonal p = 7. Substituting into the formula yields $7 \cdot q = 3 \cdot 5 + 4 \cdot 6 = 15 + 24 = 39, so q = 39 / 7 \approx 5.571. This illustrates how the theorem enables efficient geometric calculations without coordinate geometry. Another essential formula is Brahmagupta's formula for the area of a cyclic quadrilateral. Given sides a, b, c, d and semiperimeter s = (a + b + c + d)/2, the area K is

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

Named after the Indian mathematician Brahmagupta, who presented it in his 7th-century text Brahmasphutasiddhanta, this formula generalizes Heron's formula for triangles and yields the maximum possible area for given side lengths among all quadrilaterals.^[18] The law of sines extends naturally to cyclic quadrilaterals due to their inscription in a circle of radius R. For each side, the length is twice the circumradius times the sine of the opposite angle: a / \sin A = b / \sin B = c / \sin C = d / \sin D = 2R. This follows from applying the standard law of sines to the triangles formed by the diagonals, leveraging the shared circumcircle.

Cyclic Polygons

Generalization to n Points

A polygon with n \geq 3 vertices is cyclic if all its vertices lie on a single circle in the plane, which is equivalent to the existence of a unique circumcircle passing through every vertex.^[19] This property ensures that the polygon can be inscribed in the circle, generalizing the notion of concyclicity from sets of four points to arbitrary finite collections forming a simple closed chain. A necessary and sufficient condition for the vertices of an n-gon to be concyclic is that the coordinates (x_i, y_i) of all n points satisfy the general equation of a circle, x^2 + y^2 + Dx + Ey + F = 0, for some constants D, E, and F. For even n \geq 4, an additional geometric condition holds: the sums of every other interior angle (alternating interior angles) are equal, providing a verifiable test without solving for the circle parameters.^[20] These conditions distinguish cyclic polygons from non-cyclic ones, where no such encircling circle exists. All regular n-gons are cyclic, as their equal side lengths and equal interior angles allow inscription in a circle whose center coincides with the polygon's centroid. For instance, a regular pentagon is cyclic by construction, with each vertex equidistant from the center. In the case of a cyclic pentagon (n=5), the vertices must satisfy specific angle conditions, such as any three consecutive vertices subtending an arc of less than $180^\circ at the circumcenter, ensuring the polygon remains convex and properly inscribed without self-intersection.^[19]

Special Configurations and Constants

In the context of complete quadrilaterals, a special configuration arises involving the Miquel point. A complete quadrilateral consists of four lines in general position, no three concurrent, forming six intersection points and four triangles, each defined by three of the lines. The circumcircles of these four triangles concur at a single point called the Miquel point, demonstrating a concurrency that underscores the underlying concyclic relationships among the intersection points and vertices. This point serves as the center of spiral similarities mapping pairs of the quadrilateral's lines to each other.^[21] Another notable configuration linked to concyclic points is the Simson line. Given a triangle ABC and a point P on its circumcircle, the feet of the perpendiculars from P to the lines BC, CA, and AB—denoted X, Y, Z respectively—are collinear on what is known as the Simson line of P with respect to ABC. This collinearity holds precisely because P is concyclic with A, B, and C; if P is not on the circumcircle, the feet form a proper pedal triangle rather than degenerating to a line. The quadrilaterals formed by P and each pair of feet (e.g., BXPZ) are cyclic, with right angles at the feet reinforcing the circular properties.^[22] Cyclic polygons also feature intrinsic constants that quantify their efficiency in enclosing area relative to perimeter, notably the isoperimetric quotient Q = \frac{4\pi A}{P^2}, where A is the enclosed area and P is the perimeter. For regular n-gons, which are inherently cyclic, this quotient takes the explicit form Q = \frac{\pi}{n \tan(\pi/n)}, a fixed value for each n that increases toward 1 as n grows, approximating the circle's optimality. For instance, a regular hexagon yields Q \approx 0.9069, establishing a benchmark for hexagonal configurations.^[23] A representative special configuration among cyclic hexagons is one where the sums of alternating interior angles are equal, each set totaling 360 degrees—a constant relation derivable from the cyclic condition and applicable to any even-sided cyclic polygon. This property implies bilateral symmetries that facilitate the identification of additional concyclic subsets, such as points arising from diagonal intersections aligning on subsidiary circles.^[24]

Advanced Properties

Integer Side Lengths and Areas

Heronian quadrilaterals are cyclic quadrilaterals with integer side lengths and integer area. The area K of a cyclic quadrilateral with side lengths a, b, c, d is computed using Brahmagupta's formula:

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

where s = (a + b + c + d)/2 is the semiperimeter.^[18] For K to be an integer, the product (s - a)(s - b)(s - c)(s - d) must be a perfect square.^[18] Representative examples include the square with sides 5, 5, 5, 5, which has semiperimeter s = 10 and area K = 25. Another is the quadrilateral with sides 5, 6, 5, 12 (s = 14, K = 36).^[25] A further example is the isosceles trapezoid with sides 3, 3, 6, 6 (s = 9, K = 18).^[25] Primitive Heronian quadrilaterals, where the greatest common divisor of the side lengths is 1, satisfy the above conditions with coprime sides. Computational enumerations of such primitive examples have been performed up to specified perimeter limits, identifying finite lists within bounded searches (e.g., sides up to 88).^[25]

Other Geometric Properties

The power of a point theorem provides a key relation for points interacting with a circle defined by concyclic points. If two chords AB and CD intersect at an interior point P within the circle passing through concyclic points A, B, C, and D, then the products of the lengths of the chord segments are equal: PA \cdot PB = PC \cdot PD.^[26] This equality holds due to similar triangles formed by the intersecting chords and the circle's properties. The inscribed angle theorem further characterizes angles formed by concyclic points on a circle. For an inscribed angle \angle ABC subtended by arc AC at point B on the circumference, and the corresponding central angle \angle AOC at the circle's center O subtending the same arc, the measure of the inscribed angle is half the central angle: \angle ABC = \frac{1}{2} \angle AOC.^[27] This relationship arises from the isosceles triangles formed by radii to the arc endpoints and the inscribed point.^[28] For a set of concyclic points, the circumcircle—the unique circle passing through all the points—contains all points on its boundary. It serves as the minimal enclosing circle if the points do not all lie on an open semicircle; in that case, a smaller circle with the longest chord as diameter encloses them.^[29] Concyclic points appear in optimization problems, such as finding minimum Steiner trees connecting a central point to a set of points on a circle, where the tree length is bounded by geometric configurations leveraging the circle's symmetry.^[30] For instance, when points lie on a circle of radius r equidistant from a query point, the optimal tree structure exploits concyclicity to achieve efficient spanning.^[30]

Variations and Extensions

Non-Euclidean and Projective Variants

In hyperbolic geometry, the concept of concyclic points generalizes to points lying on a hyperbolic circle, horocycle, or hypercycle, depending on the configuration. For a hyperbolic triangle with side lengths a \leq b \leq c, the vertices lie on a circle if \sinh(c/2) < \sinh(a/2) + \sinh(b/2), on a horocycle if equality holds, and on a hypercycle if \sinh(c/2) > \sinh(a/2) + \sinh(b/2).^[31] A horocycle is a curve asymptotic to the boundary of the hyperbolic plane, while a hypercycle equidistant from a geodesic serves as a locus for such point sets, providing analogs to Euclidean circles for polygons whose vertices satisfy these conditions.^[31] In projective geometry, concyclicity extends to points lying on a common conic section, as circles are degenerate conics in the projective plane. Four points are concyclic if they belong to a pencil of circles, a one-parameter family of circles generated by linear combinations of two base circle equations; for instance, in an intersecting (elliptic) pencil, all circles pass through two fixed base points, ensuring any additional points on those circles share concyclicity with respect to the base pair.^[32] Cross-ratios further characterize this, preserving projective invariance for points on a conic, allowing the distinction of circular loci via harmonic properties.^[33] This framework unifies circles under projective transformations, where the two circular points at infinity—imaginary points common to all circles—facilitate the treatment of angular and metric properties projectively.^[34] Jean-Victor Poncelet advanced these ideas in the early 19th century, founding modern projective geometry through his 1822 Traité des propriétés projectives des figures, where he demonstrated that conics, including circles, retain projective properties under perspective, enabling the projection of circular problems onto simpler conic configurations.^[34] On the sphere in spherical geometry, the analog of concyclic points consists of points lying on a spherical circle, formed by the intersection of the sphere with a plane not passing through the center, generalizing planar concyclicity to curved surfaces. In three dimensions, this extends to cospherical points, where four non-coplanar points lie on a common sphere, mirroring how non-collinear points in the plane determine a circle; coplanar points are cospherical only if concyclic.^[35] Such configurations project to planar views while preserving spherical distances along great or small circles.^[36]

Computational Determination

Determining whether a set of points are concyclic computationally relies on algebraic tests for exact verification in Euclidean geometry, particularly for small numbers of points, and numerical optimization techniques for larger or noisy datasets. These methods are essential in fields requiring precise geometric analysis, such as computational geometry software and data processing pipelines. For exactly four points with coordinates (x_i, y_i) for i = 1, 2, 3, 4, concyclicity holds if and only if the determinant of the following $4 \times 4 matrix vanishes:

\begin{vmatrix} x_1 & y_1 & x_1^2 + y_1^2 & 1 \\ x_2 & y_2 & x_2^2 + y_2^2 & 1 \\ x_3 & y_3 & x_3^2 + y_3^2 & 1 \\ x_4 & y_4 & x_4^2 + y_4^2 & 1 \end{vmatrix} = 0

This condition derives from substituting the points into the general circle equation x^2 + y^2 + Dx + Ey + F = 0 and requiring linear dependence among the resulting system of equations.^[37] An alternative geometric test for four points A, B, C, D uses angles subtended by a common chord, such as AB: the points are concyclic if \angle ACB = \angle ADB (or \angle ADC = \angle ABC) when C and D lie on the same side of AB, per the converse of the inscribed angle theorem (angles in the same segment). This approach avoids coordinates but requires accurate angle computations via vector dot products or law of cosines.^[38] In cases of measurement noise or approximate concyclicity, least-squares circle fitting estimates the best-fit circle by minimizing the algebraic error \sum (x_i^2 + y_i^2 + D x_i + E y_i + F)^2. A widely adopted linear method, developed by Chernov and Ososkov, reformulates the problem as a linear system solved via singular value decomposition, achieving high numerical stability and O(n) time for n points. More recent variants, such as geometric distance minimization, enhance robustness to outliers using iterative reweighted least squares.^[39]^[40] For verifying concyclicity of n > 4 points, an efficient algorithm selects three non-collinear points to define an initial circle (via perpendicular bisector intersection), then checks the distance from the center to each remaining point against the radius, with tolerance for numerical cases; this runs in O(n) time overall. Iterative bisector checks—successively intersecting bisectors of pairs starting from an initial line—can refine the center while maintaining O(n) complexity in exact arithmetic. Numerical stability is critical here, as floating-point errors in determinants or bisector computations can lead to false negatives; robust implementations employ exact predicates or adaptive precision, as in libraries like CGAL up to 2025 releases.^[41] These techniques find applications in computer graphics for generating meshes with concyclic faces, ensuring conformal mappings and smoother discretizations in discrete exterior calculus simulations. In GPS and GNSS analysis, circle fitting to satellite-derived point clouds corrects positioning errors, improving accuracy in surveying and navigation systems.^[42]^[43]

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