Desargues' theorem | Wikipedia audio article
Channel: Subhajit Sahu
Date: 31/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Desargues%27s_theorem


00:01:13 1 History
00:01:39 2 Projective versus affine spaces
00:02:04 3 Self-duality
00:02:42 4 Proof of Desargues's theorem
00:03:12 4.1 Three-dimensional proof
00:04:41 4.2 Two-dimensional proof
00:05:10 5 Relation to Pappus's theorem
00:06:12 6 The Desargues configuration
00:07:02 7 See also



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SUMMARY
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In projective geometry, Desargues's theorem, named after Girard Desargues, states:

Two triangles are in perspective axially if and only if they are in perspective centrally.Denote the three vertices of one triangle by a, b and c, and those of the other by A, B  and C.  Axial perspectivity means that lines ab and AB meet in a point, lines ac and AC meet in a second point, and lines bc and BC meet in a third point, and that these three points all lie on a common line called the axis of perspectivity.  Central perspectivity means that the three lines Aa, Bb and Cc are concurrent, at a point called the center of perspectivity.
This intersection theorem is true in the usual Euclidean plane but special care needs to be taken in exceptional cases, as when a pair of sides are parallel, so that their "point of intersection" recedes to infinity.  Commonly, to remove these exceptions, mathematicians "complete" the Euclidean plane by "adding" points at infinity following Jean-Victor Poncelet. This results in a projective plane.
Desargues's theorem is true for the real projective plane, for any projective space defined arithmetically from a field or division ring, for any projective space of dimension unequal to two, and for any projective space in which Pappus's theorem holds. However, there are some non-Desarguesian planes in which Desargues's theorem is false.