Geodesic
Der Zusammenhang zwischen Ordnungsgraden und Kongruenzsätzen am Beispiel des Systems der Dreiecke
The Teaching of Mathematics, XIX (2016) no. 2, p. 57

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Different types of polygons are treated differently, or at least the treatments are differently focused. For triangles, congruence theorems seem to be considered to be important, while for quadrilaterals classification seems to be one of the most important aspects. Recently, we suggested a systematic treatment for quadrilaterals to substitute the often rather arbitrary-appearing common classificatory systems. For this system, we suggested certain characteristics, which lead to subsequently higher ordered quadrilaterals. This approach is generalised here and expanded to be applied also to triangles. The number of ordered triangles is significantly lower than that of quadrilaterals: there are only four, two triangles of first order and two of second order. We furthermore show that the resulting system of triangles is directly linked to the triangle congruence theorems. With this approach, we try to bridge the different ways how triangles and quadrilaterals are treated and aim at a more coherent understanding of geometry.
Classification : 97G40 G44
Keywords: classification of triangles 
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     author = {Joachim T. Haug and Carolin Haug},
     title = {Der {Zusammenhang} zwischen {Ordnungsgraden} und {Kongruenzs\"atzen} am {Beispiel} des {Systems} der {Dreiecke}},
     journal = {The Teaching of Mathematics},
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Joachim T. Haug; Carolin Haug. Der Zusammenhang zwischen Ordnungsgraden und Kongruenzsätzen am Beispiel des Systems der Dreiecke. The Teaching of Mathematics, XIX (2016) no. 2, p. 57 . http://geodesic.mathdoc.fr/item/TM2_2016_XIX_2_a0/