Desargues' Theorem

In projective geometry, Desargues' theorem, named in honor of Gérard Desargues (pronounced day ZARG), 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.

The result 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. Mathematically the most satisfying way of resolving the issue of exceptional cases is to "complete" the Euclidean plane to a projective plane by "adding" points at infinity following Poncelet.

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' theorem is false.

Read more about Desargues' TheoremHistory, Projective Versus Affine Spaces, Self-duality, Proof of Desargues' Theorem, Relation To Pappus' Theorem, The Desargues Configuration

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