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:<math>R = \sqrt{\frac{a^2b^2c^2}{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}.</math>
 
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 [[Circumscribed circleCircumcircle#Circumcircle equations|here]] and [[Circumscribed circle#Cartesian coordinates|here]].
 
=== Other concyclic points ===
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|title=On the diagonals of a cyclic quadrilateral
|url=http://forumgeom.fau.edu/FG2007volume7/FG200720.pdf |volume=7
|year=2007}}</ref><ref>{{citation |last=Hoehn |first=Larry |title=Circumradius of a cyclic quadrilateral |journal=Mathematical Gazette |volume=84 |issue=499 |date=March 2000 |pages=69–70 |doi=10.2307/3621477 |jstor=3621477}}</ref>
:<math>R=\frac{1}{4} \sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}},</math>
an expression that was derived by the Indian mathematician Vatasseri [[Parameshvara]] in the 15th century.
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\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= 311 JanuaryNovember 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>
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==Variations==
 
SomeIn authorscontexts considerwhere [[collinearlines points]]are (sets of points all belonging to a single line)taken to be a special casetype of concyclic[[generalised points,circle]] with theinfinite lineradius, being[[collinear viewedpoints]] as(points along a circlesingle ofline) infiniteare radiusconsidered to be concyclic. This point of view is helpful, for instance, when studying [[Inversivecircle geometryinversion|inversion through a circle]] andor more generally [[Möbius transformation]]s (geometric transformations generated by reflections and circle inversions), as these transformations preserve the concyclicity of points only in this extended sense.<ref>{{citation
| last = Zwikker | first = C.
| authorlink = Cornelis Zwikker
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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. 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, sides, and diagonals. For a polygon with {{mvar|n}} sides, Letlet {{math|0 < ''c''<sub>1</sub> < ... < ''c''<sub>''n''−1</sub> < +∞}} be rational numbers. These are the tangents of one quarter of the cumulative angles {{math|''θ''<sub>1</sub>}}, {{math|''θ''<sub>1</sub> + ''θ''<sub>2</sub>}}, ..., {{math|''θ''<sub>1</sub> + ... + ''θ''<sub>''n''−1</sub>}}. Let {{math|1=''q''<sub>1</sub> = ''c''<sub>1</sub>}}, andlet {{math|1=''q''<sub>''n''</sub> = 1 / ''c''<sub>''n''−1</sub>}}, and let {{math|1=''q''{{sub|''k''}} = (''c''{{sub|''k''}} − ''c''{{sub|''k''−1}}) / (1 + ''c''{{sub|''k''}}''c''{{sub|''k''−1}})}} for {{math|1=''k'' = 2, ..., ''n''−1}}. let These rational numbers are the tangents of the individual quarter angles, using the formula for the tangent of the difference of angles. Rational side lengths for the polygon circumscribed by the unit circle are thus obtained as {{math|1=''s''<sub>''k''</sub> = 4''q''<sub>''k''</sub> / (1 + ''q''<sub>''k''</sub><sup>2</sup>)}}. The rational area is {{math|1=''A'' = ∑{{sub|''k''}} 2''q''{{sub|''k''}}(1 − ''q''{{sub|''k''}}{{sup|2}}) / (1 + ''q''{{sub|''k''}}{{sup|2}}){{sup|2}}}}. These can be made into integers by scaling the side lengths by a shared constant.
<math display=block>
q_k = \frac{c_{k} - c_{k-1}}{1 + c_{k} c_{k-1}}
</math>
These rational numbers are are the tangents of the individual quarter angles, using the formula for the tangent of the difference of angles. Rational side lengths for the polygon circumscribed by the unit circle are thus obtained as {{math|1=''s''<sub>''k''</sub> = 4''q''<sub>''k''</sub> / (1 + ''q''<sub>''k''</sub><sup>2</sup>)}}. These can be made into integers by scaling the side lengths by a shared constant.
 
==Other properties==
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