Generalizing the Nagel line to circumscribed polygons by analogy and constructive defining

Michael De Villiers

doi:10.4102/pythagoras.v0i68.65

Abstract

This paper first discusses the genetic approach and the relevance of the history of mathematics for teaching, reasoning by analogy, and the role of constructive defining in the creation of new mathematical content. It then uses constructive defining to generate a new generalization of the Nagel line of a triangle to polygons circumscribed around a circle, based on an analogy between the Nagel line and the Euler line of a triangle.

Highlights

The purpose of this paper is to heuristically present a new generalisation of the Nagel line of a triangle to polygons circumscribed around a circle by making use of an interesting analogy between the Nagel line and the Euler line of a triangle (De Villiers, 2006)
In a letter to his sister, Weil (1940, p. 339) stated: “the analogies between algebraic functions and numbers have been on the minds of all the great number theorists of the time”, and in 1946, he laid the foundations of algebraic geometry from the analogy of the theory of differentiable manifolds with some constructions from algebraic topology
Myakishev (2006) provides a completely different generalisation of the Nagel line of a circumscribed quadrilateral by considering instead the centroid of a ‘perimeter’ circumscribed quadrilateral, and constructively defining a different Nagel point

Summary

Michael de Villiers

Freudenthal (1973) has criticised a deductive teaching approach, calling it a “didactical inversion” of the historical process, and that it was was anti-didactical. For example, it is possible to teach Boolean Algebra as described in De Villiers (1986a), not in the actual historical order nor from the context it originally developed, but to instead begin by focussing on the modelling of Generalizing the Nagel line to Circumscribed Polygons switching circuit problems, and only later on dealing with its axiomatization into a formal mathematical system of axioms, theorems and proofs (as well as its application to other areas such as logic, computer programming, biology, etc.). In such a case where the analogy is determined by clearly defined rules, we have a duality, as the two operations can be interchanged (as long as only these laws are involved) This duality between addition and multiplication extends to a fruitful analogy between arithmetic and geometric sequences to produce an interesting dual for the Fibonacci sequence, involving an analogous rule Tn × Tn+1 = Tn+2 for producing consecutive terms (see De Villiers, 2000).

Constructive defining

New d e fin itio n c e

The analogy between the Euler and Nagel lines of a triangle