Theorems in projective geometry

In projective geometry, theorems and principles focus on properties of geometric figures that remain invariant under projective transformations. Projective geometry is primarily concerned with relationships and properties that are not dependent on measurements of distance or angles, but rather on incidence, collinearity, and concurrency.

The Cayley–Bacharach theorem is a result in algebraic geometry that deals with the intersection of divisors on a projective space. It is particularly relevant in the study of linear systems of divisors and their properties. In its classical form, the theorem states the following: Let \( C \) be a non-singular irreducible curve of degree \( d \) in the projective plane \( \mathbb{P}^2 \).

The statement "five points determine a conic" refers to a fundamental result in projective geometry. It states that given any five points in a plane, no three of which are collinear, there exists a unique conic section (which can be an ellipse, parabola, hyperbola, or degenerate conic) that passes through all five points.

Hesse's theorem is a result in geometry that deals with the properties of projective spaces. Specifically, it states that if you have a configuration of points in a projective plane, under certain conditions, the points will lie on a conic (a curve defined by a quadratic polynomial). In a more precise sense, the theorem can be framed in terms of the collinearity of points and the conditions under which these points create a conic.

A Steiner conic, also known as a Steiner curve or a Steiner ellipse, is a specific type of conic section used in projective geometry and other areas of mathematics. It is defined in the context of a given triangle. For a triangle with vertices \( A \), \( B \), and \( C \), the Steiner conic is the unique conic that passes through the triangle's vertices and has the following additional properties: 1. Its foci are located at the triangle's centroid.

The Veblen–Young theorem is a result in set theory and topology that pertains to the structure of certain well-ordered sets and their properties. It is primarily focused on the relationship between well-ordered sets and their representations as ordinals, specifically in the context of a well-ordered set being isomorphic to an ordinal if it exhibits certain properties.