170:
20:
232:
reformed mathematical foundations in 1847 with the complete quadrangle when he noted that a "harmonic property" could be based on concomitants of the quadrangle: When each pair of opposite sides of the quadrangle intersect on a line, then the diagonals intersect the line at
198:
As systems of points and lines in which all points belong to the same number of lines and all lines contain the same number of points, the complete quadrangle and the complete quadrilateral both form
110:. For points and lines in the Euclidean plane, the diagonal points cannot lie on a single line, and the diagonals cannot have a single point of triple crossing. Due to the discovery of the
222:
of a complete quadrangle is a complete quadrilateral, and vice versa. For any two complete quadrangles, or any two complete quadrilaterals, there is a unique
102:
of the quadrangle. Similarly, among the six points of a complete quadrilateral there are three pairs of points that are not already connected by lines; the
241:. Through perspectivity and projectivity, the harmonic property is stable. Developments of modern geometry and algebra note the influence of von Staudt on
776:
584:
261:, the four lines of a complete quadrilateral must not include any pairs of parallel lines, so that every pair of lines has a crossing point.
218:), where the numbers in this notation refer to the numbers of points, lines per point, lines, and points per line of the configuration. The
459:
291:
of these same four triangles meet in a point. In addition, the three circles having the diagonals as diameters belong to a common
397:'s animation of the same results, the pencil can be hyperbolic instead of elliptic, in which case the circles do not intersect.
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838:
791:
735:
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705:
283:
to all four lines of the quadrilateral. Any three of the lines of the quadrilateral form the sides of a triangle; the
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of the four triangles formed in this way lie on a second line, perpendicular to the one through the midpoints. The
781:
812:
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570:
518:
234:
219:
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positions. The four points on the line deriving from the sides and diagonals of the quadrangle are called a
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786:
771:
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513:
267:
describes several additional properties of complete quadrilaterals that involve metric properties of the
271:, rather than being purely projective. The midpoints of the diagonals are collinear, and (as proved by
75:
223:
157:
by
Halsted. Opposite connectors cross at a codot. The configuration of the complete quadrangle is a
687:
199:
720:
393:
Wells writes incorrectly that the three circles meet in a pair of points, but, as can be seen in
299:
98:
The six lines of a complete quadrangle meet in pairs to form three additional points called the
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is a system of four lines, no three of which pass through the same point, and the six points of
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730:
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807:
675:
548:
740:
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470:
Historical Math
Monographs. See in particular tetrastigm, page 85, and tetragram, page 90.
133:
A set of contracted expressions for the parts of a complete quadrangle were introduced by
8:
750:
715:
593:
526:
394:
364:
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202:; in the notation of projective configurations, the complete quadrangle is written as (4
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122:, some authors have augmented the axioms of projective geometry with
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19:
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478:
The
Penguin Dictionary of Curious and Interesting Geometry
118:
in which the diagonal points of a complete quadrangle are
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the axis of which is the line through the orthocenters.
47:
is a system of geometric objects consisting of any four
16:
Geometric figure made of 4 points connected by 6 lines
226:
taking one of the two configurations into the other.
78:
of these lines. The complete quadrangle was called a
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of the triangles of a complete quadrilateral form a
130:
collinear, while others have been less restrictive.
475:
825:
145:. The lines of the projective space are called
23:A complete quadrangle (at left) and a complete
461:An Elementary Treatise on Modern Pure Geometry
210:) and the complete quadrilateral is written (6
153:. The "diagonal lines" of Coxeter are called
86:, and the complete quadrilateral was called a
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90:; those terms are occasionally still used.
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571:
445:
344:
137:: He calls the vertices of the quadrangle
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410:, Dover Publications, 2007 (orig. 1960).
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149:, and in the quadrangle they are called
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275:) also collinear with the center of a
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473:
464:. London, New York: Macmillan and Co.
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63:connecting the six pairs of points.
141:, and the diagonal points he calls
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451:Foundations of Projective Geometry
106:connecting these pairs are called
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855:
501:
453:. W. A. Benjamin. pp. 53–6.
736:Cremona–Richmond configuration
387:
375:
354:
338:
173:KLMN is a complete quadrangle;
1:
417:
366:Synthetic Projective Geometry
235:projective harmonic conjugate
180:projective harmonic conjugate
126:that the diagonal points are
813:Kirkman's schoolgirl problem
746:Grünbaum–Rigby configuration
531:"The Complete Quadrilateral"
93:
7:
706:Möbius–Kantor configuration
514:Encyclopedia of Mathematics
429:Projective Geometry, 2nd ed
408:Advanced Euclidean Geometry
309:
10:
860:
792:Bruck–Ryser–Chowla theorem
839:Configurations (geometry)
800:
782:Szemerédi–Trotter theorem
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686:
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224:projective transformation
200:projective configurations
772:Sylvester–Gallai theorem
458:Lachlan, Robert (1893).
331:
55:, no three of which are
844:Types of quadrilaterals
777:De Bruijn–Erdős theorem
721:Desargues configuration
509:"Quadrangle, complete"
195:
28:
808:Design of experiments
549:"Complete Quadrangle"
474:Wells, David (1991).
172:
165:Projective properties
22:
741:Kummer configuration
711:Pappus configuration
594:Incidence structures
527:Bogomolny, Alexander
482:. Penguin. pp.
253:Euclidean properties
834:Projective geometry
751:Klein configuration
731:Schläfli double six
716:Hesse configuration
696:Complete quadrangle
431:. Springer-Verlag.
406:Johnson, Roger A.,
395:Alexander Bogomolny
155:opposite connectors
45:complete quadrangle
41:projective geometry
726:Reye configuration
546:Weisstein, Eric W.
468:Cornell University
196:
39:and especially in
37:incidence geometry
35:, specifically in
29:
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447:Hartshorne, Robin
425:Coxeter, H. S. M.
293:pencil of circles
59:, and of the six
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656:Projective plane
608:Incidence matrix
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220:projective dual
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361:G. B. Halsted
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326:Quadrilateral
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801:Applications
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634:Block design
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536:Cut-the-Knot
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384:, p. 51
382:Coxeter 1987
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349:Coxeter 1987
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285:orthocenters
273:Isaac Newton
265:Wells (1991)
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124:Fano's axiom
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76:intersection
68:
44:
30:
672:Statistics
316:Newton line
247:Felix Klein
243:Mario Pieri
33:mathematics
27:(at right).
828:Categories
701:Fano plane
666:Hypergraph
466:Link from
418:References
151:connectors
112:Fano plane
80:tetrastigm
651:Incidence
554:MathWorld
519:EMS Press
159:tetrastim
147:straights
120:collinear
108:diagonals
94:Diagonals
88:tetragram
69:complete
765:Theorems
676:Blocking
646:Geometry
449:(1967).
427:(1987).
310:See also
306:system.
279:that is
521:, 2001
363:(1906)
281:tangent
257:In the
178:is the
622:Fields
490:
435:
304:coaxal
143:codots
65:Dually
49:points
484:35–36
332:Notes
277:conic
61:lines
53:plane
51:in a
756:Dual
488:ISBN
433:ISBN
298:The
245:and
190:and
139:dots
114:, a
67:, a
43:, a
182:of
128:not
82:by
31:In
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533:.
529:.
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161:.
586:e
579:t
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557:.
539:.
496:.
441:.
216:3
214:4
212:2
208:2
206:6
204:3
194:.
192:B
188:A
184:C
176:D
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