Generalizing Van Aubel Using Duality

Abstract

Michael de VilliersUniversity of Durban-Westville South AfricaA recent paper by DeTemple and Harold [1] elegantly utilized the Finsler-Hadwigertheorem to prove Van Aubel's theorem, which states that the line segments, connecting thecenters of squares constructed on the opposite sides of a quadrilateral, are congruent andlie on perpendicular lines. This result can easily be generalized by using a less well knownduality between the concepts angle and side within Euclidean plane geometry.Although similar to the general between points and lines in projectivegeometry, this duality is limited. Nevertheless, it occurs quite frequently and examplesof these are explored fairly extensively in [2]. Obviously this duality does not apply totheorems related to or based on the Fifth Postulate (compare [7]). For example, the dual tothe theorem three corresponding sides of two triangles equal imply their congruency,namely, three corresponding angles of two triangles equal imply their congruency, is notvalid. (Note, however, that the dual is perfectly true in both non-Euclidean geometries).The square is self-dual regarding these concepts as it has all angles and all sidescongruent. The parallelogram is also self-dual since it has both opposite sides andopposite angles congruent. Similarly, the rectangle and rhombus are each other's duals asshown in the table below:Rectangle RhombusAll angles congruent All sides congruentCenter equidistant from vertices, hence hascircumcircleCenter equidistant from sides, hence hasincircleAxes of symmetry bisect opposite sides Axes of symmetry bisect opposite anglesFurthermore, the congruent diagonals of the rectangle has as its dual the perpendiculardiagonals of the rhombus and is illustrated by the following two elementary results:(1) The midpoints of the sides of any quadrilateral with congruent diagonals form arhombus.