Quadrangle, complete

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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.


[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:,_complete&oldid=53686
This article was adapted from an original article by P.S. ModenovA.S. Parkhomenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article