This is a follow up to the prior post: Visualizing Exponents of Non-integer Pythagorean Generalization
For any triangle ABC where c is the long leg, an exponent exists that will satisfy an+bn=cn. Thus, via exponentiation, non-right triangles are subjected to generalization of the Pythagorean (et al.) Theorem. Normalizing c and mapping non-integer exponents to the rotating complex plane provides visualization. For n from 1 to ∞ (degenerate and isosceles respectively) of a given ρ (a:b ratio), all triangles can be simulated. Note that the locus of Pythagoras (n=2) lies in the Rex Rey plane and α represents the angle of the complex plane.
amax = ρ[0,1] = a/b
amin = ρ/ρ+1
b = a/ρ
c = (an+bn)1/n = 1
α = πn-2π
∠C = cos-1((-c2+a2+b2)/(2ab))
n = q(x) = cos-1((-c2+a2+b2)/(2ab)) ∩ f(x) = cos-1(((-(ax+bx)1/x)2+a2+b2)/(2ab))
To illustrate, I have created a Geogebra animation here: Non-Right Triangles Subjected to Pythagorean Generalization
