Editing Concyclic points (section)

===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>