Universal Affine Triangle Geometry and Four - fold

Universal Affine Triangle Geometry and Four-fold Incenter Symmetry

We develop a generalized triangle geometry, using an arbitrary bilinear form in an affine plane over a general field. By introducing standardized coordinates we find canonical forms for some basic centers and lines. Strong concurrencies formed by quadruples of lines from the Incenter hierarchy are investigated, including joins of corresponding Incenters, Gergonne, Nagel, Spieker points, Mittenp...

Universal Hyperbolic Geometry III: First Steps in Projective Triangle Geometry

We initiate a triangle geometry in the projective metrical setting, based on the purely algebraic approach of universal geometry, and yielding in particular a new form of hyperbolic triangle geometry. There are three main strands: the Orthocenter, Incenter and Circumcenter hierarchies, with the last two dual. Formulas using ortholinear coordinates are a main objective. Prominent are five partic...

Affine Maps in Triangle Geometry and Feuerbach’s Theorem

0 O ct 2 00 3 DOUBLE AFFINE HECKE ALGEBRAS OF RANK 1 AND AFFINE CUBIC SURFACES

We study the algebraic properties of the five-parameter family H(t1, t2, t3, t4; q) of double affine Hecke algebras of type C ∨ C1. This family generalizes Cherednik's double affine Hecke algebras of rank 1. It was introduced by Sahi and studied by Noumi and Stokman as an algebraic structure which controls Askey-Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is ...

The Fundamental Theorems of Elementary Geometry. an Axiomatic Analysis

Introduction. The present investigation is concerned with an axiomatic analysis of the four fundamental theorems of Euclidean geometry which assert that each of the following triplets of lines connected with a triangle is copunctual: the medians, the altitudes, the perpendicular bisectors, and the bisectors of the angles. The general framework for our discussion will be provided by an affine pl...