Universal Affine Triangle Geometry and Four-fold Incenter Symmetry
نویسندگان
چکیده
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, Mittenpunkts and the New points we introduce. The diagrams are taken from relativistic (green) geometry.
منابع مشابه
Universal Affine Triangle Geometry and Four - fold
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...
متن کاملIncenter Symmetry , Euler lines , and Schiffler Points
We look at the four-fold symmetry given by the Incenter quadrangle of a triangle, and the relation with the cirumcircle, which in this case is the nine-point conic of the quadrangle. By investigating Euler lines of Incenter triangles, we show that the classical Schiffler point extends to a set of four Schiffler points, all of which lie on the Euler line. We discover also an additional quadrangl...
متن کاملUniversal 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...
متن کاملIncenter Circles , Chromogeometry , and the Omega Triangle
Chromogeometry brings together planar Euclidean geometry, here called blue geometry, and two relativistic geometries, called red and green. We show that if a triangle has four blue Incenters and four red Incenters, then these eight points lie on a green circle, whose center is the green Orthocenter of the triangle, and similarly for the other colours. Tangents to the incenter circles yield inte...
متن کاملConic Construction of a Triangle From Its Incenter, Nine-point Center, and a Vertex
We construct a triangle given its incenter, nine-point center and a vertex by locating the circumcenter as an intersection of two rectangular hyperbolas. Some special configurations leading to solutions constructible with ruler and compass are studied. The related problem of construction of a triangle given its circumcenter, incenter, and one vertex is revisited, and it is established that such...
متن کامل