The Kepler Triangle and Its Kin

Abstract

The Kepler triangle, also known as the golden right triangle, is the right triangle with its sides of ratios '1 : φ^1/2 : φ,' where φ denotes the golden ratio. Also known are the silver right triangle and the square-root-three φ right triangle. This study introduces the generalised golden right triangle, which have sides of lengths closely related to φ and the Fibonacci numbers, F_n : '(F_{n - 2})^1/2 : φ^n/2 : (F_n)^1/2φ' for any natural number n. This formalism covers all the known φ-related right triangle, i.e., the Kepler triangle and its kin. As n tends to infinity, the ratios of the sides go to 'φ^-1 : 5^1/4 : φ.' Our model plays an important role in the classroom to study the golden ratio, the Fibonacci numbers and the Pythagorean theorem.