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 Title
 Some Construction Problems Related to the Incircle of a Triangle.
 Creator
 Bell, Amy B., Yiu, Paul Y., Florida Atlantic University
 Abstract/Description

This thesis explores several construction problems related to the incircle of a triangle. Firstly, as a generalization of a theorem of D. W. Hansen, we find two quadruples of quantities related to a triangle which have equal sums and equal sums of squares. We also study the construction problems of triangles with centroid on the incircle, and those with a specified cevian  a median, an angle bisector, or an altitude bisected by the incircle. Detailed analysis leads to designs of animation...
Show moreThis thesis explores several construction problems related to the incircle of a triangle. Firstly, as a generalization of a theorem of D. W. Hansen, we find two quadruples of quantities related to a triangle which have equal sums and equal sums of squares. We also study the construction problems of triangles with centroid on the incircle, and those with a specified cevian  a median, an angle bisector, or an altitude bisected by the incircle. Detailed analysis leads to designs of animation pictures using the dynamic software Geometer's Sketchpad.
Show less  Date Issued
 2006
 PURL
 http://purl.flvc.org/fau/fd/FA00000725
 Subject Headings
 GeometryProblems, exercises, etc, GeometryStudy and teaching (Secondary), MathematicsStudy and teachingTechnological innovations
 Format
 Document (PDF)
 Title
 The triangle of reflections.
 Creator
 Torres, Jesus, Yiu, Paul Y., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

This thesis presents some results in triangle geometry discovered using dynamic software, namely, Geometerâ€™s Sketchpad, and confirmed with computations using Mathematica 9.0. Using barycentric coordinates, we study geometric problems associated with the triangle of reflections T of a given triangle T, yielding interesting triangle centers and simple loci such as circles and conics. These lead to some new triangle centers with reasonably simple coordinates, and also new properties of some...
Show moreThis thesis presents some results in triangle geometry discovered using dynamic software, namely, Geometerâ€™s Sketchpad, and confirmed with computations using Mathematica 9.0. Using barycentric coordinates, we study geometric problems associated with the triangle of reflections T of a given triangle T, yielding interesting triangle centers and simple loci such as circles and conics. These lead to some new triangle centers with reasonably simple coordinates, and also new properties of some known, classical centers. Particularly, we show that the Parry reflection point is the common point of two triads of circles, one associated with the tangential triangle, and another with the excentral triangle. More interestingly, we show that a certain rectangular hyperbola through the vertices of T appears as the locus of the perspector of a family of triangles perspective with T, and in a different context as the locus of the orthology center of T with another family of triangles.
Show less  Date Issued
 2014
 PURL
 http://purl.flvc.org/fau/fd/FA00004167, http://purl.flvc.org/fau/fd/FA00004167
 Subject Headings
 Geometer's Sketchpad, Geometry  Study and teaching, Geometry, Hyperbolic, Mathematics  Computer network resources, Problem solving
 Format
 Document (PDF)
 Title
 The CayleyDickson algebras.
 Creator
 Khalil, Saidah Hasan, Florida Atlantic University, Yiu, Paul Y., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

This thesis studies the various effects of the nonassociativity of the CayleyDickson algebras At, t>3, especially on the structure of their automorphism groups. Beginning with the problem of composition algebra structures on euclidean spaces, we shall explain the origin of the CayleyDickson algebras, and give a selfcontained exposition on some important results on such algebras. These algebras being nonassociative, we focus on the study of the associators of the form (u,w,v) = (uw)v  u(wv...
Show moreThis thesis studies the various effects of the nonassociativity of the CayleyDickson algebras At, t>3, especially on the structure of their automorphism groups. Beginning with the problem of composition algebra structures on euclidean spaces, we shall explain the origin of the CayleyDickson algebras, and give a selfcontained exposition on some important results on such algebras. These algebras being nonassociative, we focus on the study of the associators of the form (u,w,v) = (uw)v  u(wv). The first main result, that if u and v are elements in a CayleyDickson algebra for which (u, w, v) = 0 for all w, then u and v generate a 2dimensional subalgebra isomorphic to C, was conjectured by P. Yiu, and proved by P. Eakin and A. Sathaye. We shall simplify the proof given by these latter authors. This is then used to give a simple proof of R. D. Schafer's theorem on derivations of CayleyDickson algebras, and following also Eakin and Sathaye, a proof of the conjecture by R. B. Brown on the structure of the automorphism groups of these algebras. Two simple proofs are presented for the beautiful characterization by H. Brandt that in the Cayley algebra A3 = K, conjugation by a unit element a is an automorphism if and only if a is a 6th root of unity. We shall present a geometric proof by M. A. Zorn and a purely algebraic one. The zero divisors of the CayleyDickson algebra A4 are also analyzed in detail.
Show less  Date Issued
 1993
 PURL
 http://purl.flvc.org/fcla/dt/14993
 Subject Headings
 Cayley algebras
 Format
 Document (PDF)