Often, straightforward notions from one mathematical domain, when altered even slightly, can become rich and rewarding investigations involving numerous additional domains -- particularly when the investigation includes rigorous proof. This study begins with a familiar high school geometry problem (namely finding the circumcentre of a triangle),…

The present paper describes a dynamic investigation of polygons obtained by reflecting an arbitrary point located inside or outside a given polygon through the midpoints of its sides. The activity was based on hypothesizing on the shape of the reflection polygon that would be obtained, testing the hypotheses using dynamic software, and finding a…

The importance of mathematical reasoning is unquestioned and providing opportunities for students to become involved in mathematical reasoning is paramount. The open-ended tasks presented incorporate mathematical content explored through the contexts of problem solving and reasoning. This article presents a number of simple tasks that may be…

There is a great deal of overlap between the set of practices collected under the term "computational thinking" and the mathematical habits of mind that are the focus of much mathematics instruction. Despite this overlap, the links between these two desirable educational outcomes are rarely made explicit, either in classrooms or in the…

Three unheralded results, two in Coordinate Geometry and the other in Plane Geometry, provide three proofs of a theorem dubbed by this author as "The Fundamental Theorem". The above theorem offers a uniform proof establishing the positive integral nature of the radii of the incircle and the three escribed circles associated with the right triangle…

Transformations are a central organizing idea in geometry. They are included in most geometry curricula and are likely to appear with even greater emphasis in the future, given the central role they play in the "Common Core State Standards" for K-12 mathematics. One of the attractions of geometry is the ability to draw and construct the…

In this note, a simple proof of the Generalized Ceva Theorem in plane geometry is presented. The approach is based on the principle of equilibrium in mechanics. (Contains 2 figures.)

The ancient Greek mathematicians sought to construct, by use of straight edge and compass only, all regular polygons. They had no difficulty with regular polygons having 3, 4, 5 and 6 sides, but the 7-sided heptagon eluded all their attempts. In this article, the authors discuss some cosine relations and the regular heptagon. (Contains 1 figure.)

Criteria for a reasonable axiomatic system are discussed. A discussion of the historical attempts to prove the independence of Euclids parallel postulate introduces non-Euclidean geometries. Poincare's model for a non-Euclidean geometry is defined and analyzed. (LS)

This article "opens up" the study of the relationship between quadrilaterals and circles to two sets of quadrilaterals--those that can be drawn either inside (cyclic) or outside (tangential) a circle (or both). Of course, "all" triangles are both cyclic and tangential. Or, to use the traditional Euclidean language for triangles, a unique…

Establishes that the congruence criteria for polygons with more than three sides (such as ASASA for Quadrilaterals) are easily proved within the scope of the standard high school geometry course. Also argues that elegant applications of these criteria are more easily found once these new criteria are known. (PK)

Presents an attempt to combine the history of mathematics of ancient Greece with the course on theoretical geometry taught in Greek secondary schools. Three sections present the history of ancient Greek geometry, geometrical constructions using straightedges and compasses, and an application of Ptolemy's theorem in solving ancient astronomy…