Abstract
We demonstrate a systematic, automated way of discovery of a large number of new geometry theorems on regular polygons. The applied theory includes a formula by Watkins and Zeitlin on minimal polynomials of cos frac{2pi }{n}, and a method by Recio and Vélez to discover a property in a plane geometry construction. This method exploits Wu’s idea on algebraizing the geometric setup and utilizes the theory of Gröbner bases. Also a bijective function is given that maps the investigated cases to the first natural numbers. Finally, several examples are shown that are all previously unknown results in planar Euclidean geometry.
Highlights
Obtaining interesting mathematical theorems automatically is a dream of many mathematicians
We remark that the “geometry theorems” we obtain in this article are related to lengths appearing in regular polygons
The software tool runs in a modern web browser, for example, Google Chrome 64. It uses the Giac computer algebra system to compute eliminations, and GeoGebra to visualize the obtained results on-the-fly— the results can be saved as a GeoGebra file
Summary
Obtaining interesting mathematical theorems automatically is a dream of many mathematicians. One example is Karst’s statement (Fig. 1) that claims parallelism of segments O B and J M in a regular nonagon, constructed with Mathematica in http://mathworld.wolfram.com/ RegularNonagon.html by citing [2] (see https://www.geogebra.org/m/AXd5ByHX#material/x5u93pFr for a freely available online resource). Such theorems are, difficult to find in the literature, and they seem hard to discover in a purely mechanical way. We remark that the “geometry theorems” we obtain in this article are related to lengths appearing in regular polygons These results may be considered as “algebraic theorems” because the lengths are always expressed by roots of algebraic equations. Concurrency of lines can be handled in a way that is described in Example 8
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