Difference between revisions of "Quadrangle, complete"
From Encyclopedia of Mathematics
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| − | A collection of four points | + | {{TEX|done}} |
| + | 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.). | ||
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/q076010a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/q076010a.gif" /> | ||
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Figure: q076010a | Figure: q076010a | ||
| − | The points | + | 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 | + | 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|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. |
Revision as of 16:25, 11 April 2014
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://encyclopediaofmath.org/index.php?title=Quadrangle,_complete&oldid=31508
Quadrangle, complete. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadrangle,_complete&oldid=31508
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