# Quadrangle, complete

A collection of four points $A,B,C,D$ (lying in a plane), no three of which lie on the same line, and the six lines connecting these points (cf. Fig.).

Figure: q076010a

The points $A,B,C,D$ are called the vertices, and the lines $AB,CD,AC,BD,BC,AD$ are called the edges of the complete quadrangle. Edges that have no common vertex are called opposite; the points $P,Q,R$ of intersection of the opposite edges are called diagonal points.

If $S$ and $T$ are the points of intersection of the line $PQ$ with the lines $AD$ and $BC$, then the four points $P,Q,S,T$ form a harmonic quadruple of points. The dual figure to a quadrangle is called a quadrilateral — a collection of four lines (in a plane), no three of which contain a common point.

#### Comments

#### References

[a1] | H.S.M. Coxeter, "Projective geometry" , Springer (1987) pp. 7; 95 |

[a2] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963) |

[a3] | M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French) |

**How to Cite This Entry:**

Quadrangle, complete.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Quadrangle,_complete&oldid=31508