Editing Concyclic points (section)

== Cyclic polygons ==
[[File:annuli_with_same_area_around_unit_regular_polygons.svg|thumb|As a corollary of the [[annulus (mathematics)|annulus]] chord formula, the&nbsp;area bounded by the [[circumcircle]] and [[incircle]] of every unit regular {{mvar|n}}-gon is {{pi}}/4]]

More generally, a [[polygon]] in which all vertices are concyclic is called a ''cyclic polygon''. A polygon is cyclic if and only if the perpendicular bisectors of its edges are [[concurrent lines|concurrent]].<ref>{{citation
 | last1 = Byer | first1 = Owen
 | last2 = Lazebnik | first2 = Felix
 | last3 = Smeltzer | first3 = Deirdre L. | author3-link = Deirdre Smeltzer
 | isbn = 9780883857632
 | page = 77
 | publisher = Mathematical Association of America
 | title = Methods for Euclidean Geometry
 | url = https://books.google.com/books?id=W4acIu4qZvoC&pg=PA77
 | year = 2010}}.</ref> Every [[regular polygon]] is a cyclic polygon.

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 {{nowrap|1, 3, 5, …}} are equal, and sides {{nowrap|2, 4, 6, …}} are equal).<ref>{{cite journal|last=De Villiers|first=Michael|title=95.14 Equiangular cyclic and equilateral circumscribed polygons|journal=[[The Mathematical Gazette]]|volume=95|issue= 532 |date=March 2011|pages=102–107|doi=10.1017/S0025557200002461|jstor= 23248632|s2cid=233361080 }}</ref>

A cyclic [[pentagon]] with [[rational number|rational]] sides and area is known as a [[Robbins pentagon]]. In all known cases, its diagonals also have rational lengths, though whether this is true for all possible Robbins pentagons is an unsolved problem.<ref>{{cite journal|last1=Buchholz|first1=Ralph H.|last2=MacDougall|first2=James A.|doi=10.1016/j.jnt.2007.05.005|issue=1|journal=[[Journal of Number Theory]]|mr=2382768|pages=17–48|title=Cyclic polygons with rational sides and area|volume=128|year=2008|doi-access=free}}</ref>

In any cyclic {{mvar|n}}-gon with even {{mvar|n}}, the sum of one set of alternate angles (the first, third, fifth, etc.) equals the sum of the other set of alternate angles. This can be proven by induction from the {{math|1=''n'' = 4}} case, in each case replacing a side with three more sides and noting that these three new sides together with the old side form a quadrilateral which itself has this property; the alternate angles of the latter quadrilateral represent the additions to the alternate angle sums of the previous {{mvar|n}}-gon.

A [[tangential polygon]] is one having an [[inscribed circle]] tangent to each side of the polygon; these tangency points are thus concyclic on the inscribed circle. Let one {{mvar|n}}-gon be inscribed in a circle, and let another {{mvar|n}}-gon be tangential to that circle at the vertices of the first {{mvar|n}}-gon. Then from any point {{mvar|P}} on the circle, the product of the perpendicular distances from {{mvar|P}} to the sides of the first {{mvar|n}}-gon equals the product of the perpendicular distances from {{mvar|P}} to the sides of the second {{mvar|n}}-gon.<ref>{{cite book |first=Roger A. |last=Johnson |title=Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle |publisher=Houghton Mifflin Co. |year=1929 |hdl=2027/wu.89043163211 |page=72}} Republished by Dover Publications as ''Advanced Euclidean Geometry'', 1960 and 2007.</ref>

===Point on the circumcircle===

Let a cyclic {{mvar|n}}-gon have vertices {{math|''A''{{sub|1}} , …, ''A{{sub|n}}''}} on the unit circle. Then for any point {{mvar|M}} on the minor arc {{math|''A''{{sub|1}}''A{{sub|n}}''}}, the distances from {{mvar|M}} to the vertices satisfy<ref>{{cite web|title=Inequalities proposed in ''Crux Mathematicorum''|work=The IMO Compendium|url=http://www.imomath.com/othercomp/Journ/ineq.pdf|at=p. 190, #332.10}}</ref>

:<math>\begin{cases}
  \overline{MA_1} + \overline{MA_3} + \cdots + \overline{MA_{n-2}} + \overline{MA_n} < n/\sqrt{2} & \text{if } n \text{ is odd}; \\
  \overline{MA_1} + \overline{MA_3} + \cdots + \overline{MA_{n-3}} + \overline{MA_{n-1}} \leq n/\sqrt{2} & \text{if } n \text{ is even}.
\end{cases}</math>

For a regular {{mvar|n}}-gon, if <math>\overline{MA_i}</math> are the distances from any point {{mvar|M}} on the circumcircle to the vertices {{mvar|A{{sub|i}}}}, then <ref name=Mamuka >{{cite journal| last1= Meskhishvili |first1= Mamuka| date=2020|title=Cyclic Averages of Regular Polygons and Platonic Solids |journal= Communications in Mathematics and Applications|volume=11|pages=335–355|doi= 10.26713/cma.v11i3.1420|doi-broken-date= 1 November 2024|arxiv= 2010.12340|url= https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065}}</ref>

:<math>3(\overline{MA_1}^2 + \overline{MA_2}^2 + \dots +  \overline{MA_n}^2)^2=2n (\overline{MA_1}^4 + \overline{MA_2}^4 + \dots + \overline{MA_n}^4).</math>

===Polygon circumscribing constant===
[[File:Kepler constant inverse.svg|thumb|right|upright=0.8|A sequence of circumscribed polygons and circles.]]

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''

:<math>\prod_{n=3}^\infty \frac 1 {\cos\left( \frac\pi n \right)} = 8.7000366\ldots.</math>

{{OEIS|A051762}}. The reciprocal of this constant is the [[Kepler–Bouwkamp constant]].