Hinge theorem

Abstract: The asymmetry of the relativistic addition law for noncollinear velocities under the velocity permutation leads to two modified triangles on a Euclidean plane depicting the addition of unpermuted and permuted velocities and the appearance of a nonzero angle ω between two resulting velocities. A particle spin rotates through the same angle ω under a Lorentz boost with a velocity noncollinear to the particle velocity. Three mutually connected three-parameter representations of the angle ω, obtained by the author earlier, express the three-parameter symmetry of the sides and angles of two Euclidean triangles identical to the sine and cosine theorems for the sides and angles of a single geodesic triangle on the surface of a pseudosphere. Namely, all three representations of the angle ω, after a transformation of one of them, coincide with the representations of the area of a pseudospherical triangle expressed in terms of any two of its sides and the angle between them. The angle ω is also symmetrically expressed in terms of three angles or three sides of a geodesic triangle, and therefore it is an invariant of the group of triangle motions over the pseudo-sphere surface, the group that includes the Lorentz group. Although the pseudospheres in Euclidean and pseudo-Euclidean spaces are locally isometric, only the latter is isometric to the entire Lobachevsky plane and forms a homogeneous isotropic curved 4-velocity space in the flat Minkowski space. In this connection, relativistic physical processes that may be related to the pseudosphere in Euclidean space are especially interesting.

Abstract: Preface. Introduction. About the Author. Chapter 1: Elementary Euclidean Geometry Revisited. Review of Basic Concepts of Geometry. Learning from Geometric Fallacies. Common Nomenclature. Chapter 2: Concurrency of Lines in a Triangle. Introduction. Ceva's Theorem. Applications of Ceva's Theorem. The Gergonne Point. Chapter 3: Collinearity of Points. Duality. Menelaus's Theorem. Applications of Menelaus's Theorem. Desargues's Theorem. Pascal's Theorem. Brianchon's Theorem. Pappus's Theorem. The Simson Line. Radical Axes. Chapter 4: Some Symmetric Points in a Triangle. Introduction. Equiangular Point. A Property of Equilateral Triangles. A Minimum Distance Point. Chapter 5: More Triangle Properties. Introduction. Angle Bisectors. Stewart's Theorem. Miquel's Theorem. Medians. Chapter 6: Quadrilaterals. Centers of a Quadrilateral. Cyclic Quadrilaterals. Ptolemy's Theorem. Applications of Ptolemy's Theorem. Chapter 7: Equicircles. Points of Tangency. Equiradii. Chapter 8: The Nine-Point Circle. Introduction to the Nine-Point Circle. Altitudes. The Nine-Point Circle Revisited. Chapter 9: Triangle Constructions. Introduction. Selected Constructions. Chapter 10: Circle Constructions. Introduction. The Problem of Apollonius. Chapter 11: The Golden Section and Fibonacci Numbers. The Golden Ratio. Fibonacci Numbers. Lucas Numbers. Fibonacci Numbers and Lucas Numbers in Geometry. The Golden Rectangle Revisited. The Golden Triangle. Index.

Abstract: A designer's triangle providing for the selection of angles from 15 to 90 degrees in a single implement, in increments of 15 degrees or less, and further including provisions for draft or taper angles to vary the aforesaid angles. Further embodiments include metric, fractional and tenth of inch embossed scales, protection against smudging, and marked or designated radii at a plurality of points on the triangle.