nLab Pappus' involution theorem

Idea

Given a complete quadrangle and a line $p$ not through a vertex of the quadrangle and in the same plane as the quadrangle, consider the 6 points of intersection of sides of the quadrangle with the line $p$.

Pappus’ involution theorem is a result in projective geometry stating that in the above situation there is a projective involution (a projective automorphism which is also an involution) of the line $p$ which interchanges each of the intersections of a side of the quadrangle with $p$ with the intersection of the opposite side of the quadrangle with $p$.

This morphism of projective spaces is not a perspectivity but a composition of two perspectivities.

References

• John Bamberg, Tim Penttila, Analytic projective geometry, Cambridge University Press 2023 doi

• Ruben Vigara: An application of Pappus’ Involution Theorem in euclidean and non-euclidean geometry [arXiv:1412.7414]

category: geometry