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The hyperbolic Pythagorean theorem in the Poincaré disc model of hyperbolic geometry. (English) Zbl 1004.51025

The author considers triangles \(T\) in a hyperbolic plane \(\pi\). Using a “Möbius-addition” \(\oplus\) in \(\pi\) based on a (special) hyperbolic distance function it can be shown that, for a right-angled triangle with “cathetes” \(A,B\) and “hypothenuse” \(C\) Pythagoras’ theorem remains valid: \(|A|^2 \oplus |B|^2 = |C|^2\).
Remark: This model independent statement is proved in the Poincaré disc model of \(\pi\) represented by the euclidean unit disc as usual.

MSC:

51M09 Elementary problems in hyperbolic and elliptic geometries
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