During my teaching career, I always tried to follow the example of mathematics teachers like Tiekie de Jager from Rondebosch Boys' High, and the inspiring writings of George Polya, to encourage my students to think creatively and to experiment with their own ideas. So it was always exciting to have students, every now and again, come up with some novel ideas. The ideas didn't have to be ground-breaking to be appreciated, and gave students self-confidence in their own abilities. Here is just one example of such an experience.
In August 2012, I challenged my class of prospective mathematics teachers at UKZN to construct a 'dynamic equilateral triangle' using dynamic geometry software, i.e. a triangle which would remain equilateral no matter how you dragged the vertices. Of course, what will probably immediately come to mind for the experienced geometer is the famous construction of an equilateral triangle using a compass given in Proposition 1 of Euclid's Elements.
But much to my surprise and delight, one of my students, Anele Moli, came up with the novel construction below.
Step 1: Construct a circle with centre A and radius AB. (Already done in sketch below).
Step 2 Click on the 'Construct perpendicular bisector of AB' button.
Step 3 Click on the 'Complete Triangle ABC' button to show the completed equilateral triangle ABC.
Anele's Method: Constructing an equilateral triangle
Drag A or B to check if it remains equilateral.
Challenge
Can you explain (prove) why Anele's method works?
If stuck, try this Hint.
Related Links
A Fibonacci Generalization - Kendal's theorem
Sylvie's Theorem
The quasi-circumcentre (Renata's theorem) and quasi-incentre of a quadrilateral
Clough's Theorem (a variation of Viviani)
2D Generalizations of Viviani's Theorem
Parallelogram Distances
Napoleon's Theorem: Generalizations, Variations & Converses
The 120 degree Rhombus (or Conjoined Twin Equilateral Triangles) Theorem
Dao Than Oai’s generalization of Napoleon’s theorem
Jha and Savaran’s generalisation of Napoleon’s theorem
Fermat-Torricelli Point Generalization (aka Jacobi's theorem)
Some Circle Concurrency Theorems
A generalization of the Cyclic Quadrilateral Angle Sum theorem
Dirk Laurie Tribute Problem
Cross's (Vecten's) theorem & generalizations to quadrilaterals
Matric Exam Geometry Problem - 1949
SA Mathematics Olympiad Problem 2016, Round 1, Question 20
Intersecting Circles Investigation
External Links
SA Mathematics Olympiad
Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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First created with JavaSketchpad by Michael de Villiers, 30 August 2012; updated to WebSketchpad, 19 October 2024.