Kepler Triangle
A Kepler triangle is named after Johannes Kepler, is a right-angled triangle whose sides are in geometric progression. If the common ratio of this geometric progression is √x, then the sides are in the ratio 1 : √x : x, so we have 1 + x = x2. This means that x = ϕ [= (1 + √5)/2]. So the sides of the Kepler triangle are in the ratio 1 : √ϕ : ϕ ≈ 1 : 1.272 : 1.618.
Clearly, a triangle with sides k, k√ϕ, kϕ (k > 0) is also a Kepler triangle since
ϕ2 = ϕ + 1 ⟹ (kϕ)2 = (k√ϕ)2 + k2.
Note that a triangle that is similar to a Kepler triangle is itself a Kepler triangle.
