Kepler triangle

Kepler's triangle is a term used in triangle geometry . A right-angled triangle of the Euclidean plane is referred to as such , the three side lengths of which form a finite geometric sequence , that is to say that its side lengths are in relation to one another. With regard to the terminology - following the sentence given below - reference is made to a remark made by Johannes Kepler , according to which geometry possesses two treasures , namely on the one hand the Pythagorean theorem and on the other hand the subdivision of a route according to the golden ratio . ${\displaystyle 1:{\sqrt {A}}:A}$

Phrase on the connection with the golden ratio

From the Pythagorean theorem it follows that the relationship of the geometric sequence is the condition ${\displaystyle A}$

${\displaystyle 1+A=A^{2}}$

must meet. This is exactly the definition equation for the division ratio of the golden section , it is therefore with ${\displaystyle \Phi }$ ${\displaystyle A=\Phi }$

${\displaystyle \Phi ={\frac {1+{\sqrt {5}}}{2}}}$.

The sentence follows:

A right triangle in the Euclidean plane is a Kepler triangle if and only if it is similar to a triangle with the lengths of its sides . ${\displaystyle 1\;,\;{\sqrt {\Phi }}\;,\;\Phi }$

Individual evidence

1. ↑ ^{a b} Claudi Alsina, Roger B. Nelsen: Enchanting evidence: a journey through the elegance of mathematics. 2013, pp. 88-89
2. ^ R. Herz-Fischler: A “very pleasant theorem” . In: College Mathematics Journal . tape 24 , 1993, pp. 318-324 .