Andrei Moldavanov, Classical
right-angled triangles and the golden ratio,
Forum Geometricorum, 17 (2017) 433--437.
Abstract. In this article, we consider the family of classical right-angle triangles in 2-dimensional Euclidean space. We consider triangle with an arbitrary leg ratio k and show that at , where p = ±1, the area of all built-in triangles is linked to each other by the golden ratio φ. Keeping , we address changes in above triangles occurring at the planar similarity transformation, prove an invariancy of the area ratio between predecessor and successor triangle and show that evolution curve is a logarithmic spiral. Reason of such geometrical features is associated with the unique nature of φ providing parity between the linear and non-linear properties of geometry objects.
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