concyclic points

perpendicular bisectors of chords between concyclic points In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. A polygon whose vertices are concyclic is called a cyclic polygon, and the circle is called its circumscribing circle or circumcircle. All concyclic points are equidistant from the center of the circle. Three points in the plane that do not all fall on a straight line are concyclic, so every triangle is a cyclic polygon, with a well-defined circumcircle. However, four or more points in the plane are not necessarily concyclic. After triangles, the special case of cyclic quadrilaterals has been most extensively studied.

perpendicular bisectors of chords between concyclic points In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. A polygon whose vertices are concyclic is called a cyclic polygon, and the circle is called its circumscribing circle or circumcircle. All concyclic points are equidistant from the center of the circle. Three points in the plane that do not all fall on a straight line are concyclic, so every triangle is a cyclic polygon, with a well-defined circumcircle. However, four or more points in the plane are not necessarily concyclic. After triangles, the special case of cyclic quadrilaterals has been most extensively studied. R = \sqrt{\frac{a^2b^2c^2}{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}. The equation of the circumcircle of a triangle, and expressions for the radius and the coordinates of the circle's center, in terms of the Cartesian coordinates of the vertices are given here. R=\frac{1}{4} \sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}, an expression that was derived by the Indian mathematician Vatasseri Parameshvara in the 15th century. By Ptolemy's theorem, if a quadrilateral is given by the pairwise distances between its four vertices , , , and in order, then it is cyclic if and only if the product of the diagonals equals the sum of the products of opposite sides: AC \cdot BD = AB \cdot CD + BC \cdot AD. If two lines, one containing segment and the other containing segment , intersect at , then the four points , , , are concyclic if and only if AX \cdot XC = BX \cdot XD. The intersection may be internal or external to the circle. This theorem is known as power of a point. A convex quadrilateral is orthodiagonal (has perpendicular diagonals) if and only if the midpoints of the sides and the feet of the four altitudes are eight concyclic points, on what is called the eight-point circle.

Point on the circumcircle

Let a cyclic -gon have vertices on the unit circle. Then for any point on the minor arc , the distances from to the vertices satisfy \begin{align} MA_1 + MA_3 + \cdots + MA_{n-2} + MA_n & For a regular -gon, if are the distances from any point on the circumcircle to the vertices , then 3 \left(MA_1^2 + MA_2^2 + \dots + MA_n^2 \right)^2 = 2n \left(MA_1^4 + MA_2^4 + \dots + MA_n^4\right).

Polygon circumscribing constant

Any regular polygon is cyclic. Consider a unit circle, then circumscribe a regular triangle such that each side touches the circle. Circumscribe a circle, then circumscribe a square. Again circumscribe a circle, then circumscribe a regular pentagon, and so on. The radii of the circumscribed circles converge to the so-called polygon circumscribing constant \prod_{n=3}^\infty \frac 1 {\cos \frac\pi n} = 8.7000366\ldots. . The reciprocal of this constant is the Kepler–Bouwkamp constant.

Variations

In contexts where lines are taken to be a type of generalised circle with infinite radius, collinear points (points along a single line) are considered to be concyclic. This point of view is helpful, for instance, when studying inversion through a circle or more generally Möbius transformations (geometric transformations generated by reflections and circle inversions), as these transformations preserve the concyclicity of points only in this extended sense. In the complex plane (formed by viewing the real and imaginary parts of a complex number as the x and y Cartesian coordinates of the plane), concyclicity has a particularly simple formulation: four points in the complex plane are either concyclic or collinear if and only if their cross-ratio is a real number.

Integer area and side lengths

Some cyclic polygons have the property that their area and all of their side lengths are positive integers. Triangles with this property are called Heronian triangles; cyclic quadrilaterals with this property (and that the diagonals that connect opposite vertices have integer length) are called Brahmagupta quadrilaterals; cyclic pentagons with this property are called Robbins pentagons. More generally, versions of these cyclic polygons scaled by a rational number will have area and side lengths that are rational numbers. Let be the angle spanned by one side of the cyclic polygon as viewed from the center of the circumscribing circle. Similarly define the central angles for the remaining sides. Every Heronian triangle and every Brahmagupta quadrilateral has a rational value for the tangent of the quarter angle, , for every value of . Every known Robbins pentagon (has diagonals that have rational length and) has this property, though it is an unsolved problem whether every possible Robbins pentagon has this property. The reverse is true for all cyclic polygons with any number of sides; if all such central angles have rational tangents for their quarter angles then the implied cyclic polygon circumscribed by the unit circle will simultaneously have rational side lengths and rational area. Additionally, each diagonal that connects two vertices, whether or not the two vertices are adjacent, will have a rational length, as will each apothem and each sagitta. Such a cyclic polygon can be scaled so that its area and lengths are all integers. This reverse relationship gives a way to generate cyclic polygons with integer area and lengths. For a polygon with sides, let be nonzero rational numbers where . These are the tangents of one quarter of the cumulative angles , , ..., : \begin{align} c_1 &= \tan \tfrac14 \theta_1 \\ c_2 &= \tan \tfrac{1}{4}(\theta_1 + \theta_2) \\ \vdots \ \, & \\ c_{n-1} &= \tan \tfrac14(\theta_1 + \dots + \theta_{n-1} ) \end{align} Using the formula for the tangent of the difference of angles, the tangents of the individual quarter angles can be found as \begin{align} q_1 &= c_1, \\[8pt] q_k &= \frac{c_{k} - c_{k-1}}{1 + c_{k}c_{k-1}}, \quad k = 2, \ldots, n-1 \\[4pt] q_n &= \frac{1}{c_{n-1}}. \end{align} From these angle tangents , the lengths of the chords forming the sides of the circumscribed polygon can thus be found using the half-angle identity: s_k = 2 \sin \frac{\theta_k}{2} = \frac{4q_k}{1 + q_k^2}, thereby giving the side lengths of the circumscribed polygon. The rational area is found by summing the areas of all triangles formed by the radii and chords (note that the radii are of length 1): A = \sum_k \sin \frac{\theta_k}{2} \cos \frac{\theta_k}{2} = \sum_k \frac{2q_k(1 - q_k^2)}{(1 + q_k^2)^2}. These can be made into integers by scaling the side lengths by a shared constant. This approach generates all cyclic polygons with integer area and side lengths when they also have only rational diagonals, but it will not generate any with an irrational diagonal; thus, it is an unsolved problem whether it generates all Robins pentagons.

Other properties

A set of five or more points is concyclic if and only if every four-point subset is concyclic. This property can be thought of as an analogue for concyclicity of the Helly property of convex sets.

References

Text derived from https://en.wikipedia.org/w/index.php?oldid=1376530199 under CC BY-SA 4.0; modifications have been made.