Geometry--Study and teaching (Secondary)

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.

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.