CTK Insights

04 Apr

Weekly report, the week of March 28, 2016

The week started with Gregoire Nicollier's posting where he applied his beautiful theory of spectral decomposition of polygons to quadrilaterals. I added a GeoGebra illustration to make the theory more accessible, not that it needed that.
Spectral decomposition of quadrilaterals

Gregoire showed how his theory supplied one-line proofs to the problems considered in the previous week: squares on the sides of a parallelogram and squares on the sides of an arbitrary quadrilateral. The whole episode serves a signal example of the unifying, illuminating power of general theory.

Next came a posting about emergence of a parallelogram in a trapezoid, with one sides passing through the apex of an isosceles triangle with the base the opposite side of the trapezoid. The problem, especially when furnished with an illustration, appeared quite intuitively simple. However, several remarks made online by the visitors showed that the intuition alone may be quite misleading.

Parallelogram in trapezoid

One solution employed analytic geometry, another disguised indirect reasoning based on the uniqueness of the solution to the famous Heron's problem. But the third solution stemmed from the observation (by Gregoire Nicollier) that the whole configuration is just a special case of Pappus' theorem.

On April 1st I indulged myself in a doodling with a problem that appeared to beg for a generalization and seemed to hold a promise of a non-trivial result. I am afraid that the promise remained illusionary. But the weekend brought a very satisfying compensation.

Leo Giugiuc and Dan Sitaru shared on the CutTheKnotMath facebok page an elementary problem from their article at the Gazeta Matematica.

An inequality from Gazeta Matematica

The purpose of the article was to introduce an application of Linear Algebra to proving various inequalities. The one they posted at the facebook was the simplest example of such an application. The posting engendered a stream of responses with elementary solutions. As of this writing, ten different proofs have been added to their original one. The present collection supplies a perfect illustration to the fact that even the most simplest of the problems may be looked at from various angles. I wish that the authors of school textbooks that often include only answers or "solutions to the odd-numbered problems" paid more attention to the possibility of alternative view points.

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