# Desargues' theorem

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## Desargues' theorem

[dā′zärgz ‚thir·əm]
(mathematics)
If the three lines passing through corresponding vertices of two triangles are concurrent, then the intersections of the three pairs of corresponding sides lie on a straight line, and conversely.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Suppose that M is embedded into a Desarguesian projective space B.
Neither [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be embedded into a Desarguesian projective space.
which can be realized in a (Desarguesian) projective space seems natural.
As a rule, in the sequel we consider only embeddings into Desarguesian spaces.
* Show there exist semifield planes of order [2.sup.r], for any odd integer r that admit Desarguesian subplanes of order [2.sup.2].
To illustrate the complexity of the situation, recall again that there are semifield planes of order 25 that contain semifield subplanes of order [2.sup.2], necessarily Desarguesian and there are semifields planes of orders [2.sup.5k] or [2.sup.7k], for k odd, that admit Desarguesian subplanes of order 22.
(2) The transposed then dualized commutative binary Knuth semifield planes of order [2.sup.n] for n = 5k, or 7k, for k odd, are symplectic and admit Desarguesian subplanes of orders [2.sup.2].
Every commutative binary Knuth semifield plane of order [2.sup.5k] or [2.sup.7k], for k odd, admits a Desarguesian subplane of order 4.
So there exist precisely eight extensions of [L.sub.4] to a Desarguesian spread of 85 lines in PG(7,2).
Already in , [section] 20 and  it was established that in the 3-dimensional betweenness space every bundle of lines through a fixed point has the structure of a Desarguesian projective plane (see also , Sec.
Every Moufang-type betweenness plane is Desarguesian.
Desarguesian Projective Planes [PG2].This family contains the point-line graphs of Desarguesian projective planes PG2(q).

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