Math 2.0

Geometry Definition

Cyclic quadrilateral

Builds on Concyclic pointsCircumcircle

Definition

Let ABCDABCD be a quadrilateral in the Euclidean plane ℝ2\mathbb{R}^2, with the vertices AA, BB, CC, DD in this order. The quadrilateral ABCDABCD is cyclic if its four vertices are concyclic. The circle through them is its circumcircle.

Remarks

A convex quadrilateral ABCDABCD is cyclic if and only if its opposite angles add up to π\pi: ∠DAB+∠BCD=π\angle DAB + \angle BCD = \pi. The direct part is Proposition 22 of Book III of Euclid's Elements. Each rectangle is cyclic. A rhombus that is not a square is not cyclic. Usually the quadrilateral is taken to be convex. A crossed quadrilateral with concyclic vertices is a crossed cyclic quadrilateral.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph
Circumcircle, concept, no evidence levelCircumcircleconceptConcyclic points, concept, no evidence levelConcyclic pointsconceptCyclic quadrilateral, definition, no evidence levelCyclic quadrilateraldefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Circumcircle concept
  2. Concyclic points concept
  3. Cyclic quadrilateral definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Cyclic_quadrilateral, lead section
  2. Web https://en.wikipedia.org/wiki/Cyclic_quadrilateral, Supplementary angles
About this page
Tag0CMC2KAXFS
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WritingW1 stated
MSC51M04
DefinesCyclic quadrilateral
UsesConcyclic points, Circumcircle

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