Yiu, Paul Y.

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 pictures using the dynamic software Geometer's Sketchpad.

This thesis presents some results in triangle geometry discovered using dynamic
software, namely, Geometer’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.

This thesis studies the various effects of the nonassociativity of the Cayley-Dickson 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 Cayley-Dickson algebras, and give a self-contained 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 Cayley-Dickson algebra for which (u, w, v) = 0 for all w, then u and v generate a 2-dimensional 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 Cayley-Dickson 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 Cayley-Dickson algebra A4 are also analyzed in detail.

As an educator, my greatest concern is to provide my students with instruction that will raise their level of understanding in mathematics. For geometry in particular, the van Hiele Theory is a way to measure a student's level of geometric understanding. Geometry instruction that raises a student's van Hiele level can be enhanced with two important resources, the ancient text of Euclid's Elements a contemporary dynamic geometry software program like the Geometer's Sketchpad. Euclid's Elements can be read as a book of geometric constructions rather than a list of theorems neatly arranged in logical order. The Geometer's Sketchpad is a convenient and efficient tool for geometric constructions. It is only natural to incorporate these two resources in geometry instruction. The logical structure of Euclid's Elements is intimidating to most learners, but teaching and learning need not be pursued logically linearly. This thesis is an attempt to incorporate some of the important constructions in Euclid's Elements with Geometer's Sketchpad, through the design of instruction modules in geometric constructions, to help students better understand geometry, and to improve their van Hiele level of understanding of geometry.