Editing Concyclic points (section)

== Cyclic quadrilaterals ==
{{main|Cyclic quadrilateral}}

[[File:Four concyclic points.png|thumb|Four concyclic points forming a [[cyclic quadrilateral]], showing two equal angles]]

A quadrilateral ''ABCD'' with concyclic vertices is called a [[cyclic quadrilateral]]; this happens [[if and only if]] <math>\angle CAD = \angle CBD</math> (the  [[inscribed angle theorem]]) which is true if and only if the opposite angles inside the quadrilateral are [[supplementary angle|supplementary]].<ref>{{citation
 | last = Pedoe | first = Dan
 | edition = 2nd
 | isbn = 9780883855188
 | page = xxii
 | publisher = Cambridge University Press
 | series = MAA Spectrum
 | title = Circles: A Mathematical View
 | url = https://books.google.com/books?id=rlbQTxbutA4C&pg=PR22
 | year = 1997}}.</ref> A cyclic quadrilateral with successive sides ''a'', ''b'', ''c'', ''d'' and [[semiperimeter]] ''s'' = (''a'' + ''b'' + ''c'' + ''d'')&thinsp;/&thinsp;2 has its circumradius given by<ref name=Alsina2>{{citation
|last1=Alsina |first1=Claudi |last2=Nelsen |first2=Roger B.
|journal=Forum Geometricorum
|pages=147–9
|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.

By [[Ptolemy's theorem]], if a quadrilateral is given by the pairwise distances between its four vertices ''A'', ''B'', ''C'', and ''D'' in order, then it is cyclic if and only if the product of the diagonals equals the sum of the products of opposite sides: 

: <math>AC \cdot BD = AB \cdot  CD + BC \cdot AD.</math>

If two lines, one  containing segment ''AC'' and the other containing segment ''BD'', intersect at ''X'', then the four points ''A'', ''B'', ''C'', ''D'' are concyclic if and only if<ref>{{citation |last=Bradley |first=Christopher J. |title=The Algebra of Geometry: Cartesian, Areal and Projective Co-Ordinates |publisher=Highperception |year=2007 |isbn=978-1906338008 |page=179 |oclc=213434422}}</ref>
:<math>\displaystyle AX\cdot XC = BX\cdot XD.</math>

The intersection ''X'' 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 [[Quadrilateral#Special line segments|altitudes]] are eight concyclic points, on what is called the '''eight-point circle'''.