Non-Right Triangles Subjected to Pythagorean Generalization | Steve Wait – July 7, 2023

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 Re_x Re_y plane and α represents the angle of the complex plane.

 

${a}_{max}=\rho\left[0,1\right]=\frac{{a}}{b}$

 

${a}_{min}=\frac{\rho}{\rho+1}$

 

$b=\frac{{a}}{\rho}$

 

$c=\sqrt[{n}]{{a}^n+{b}^n}=1$

 

(angle of complex plane) = $α=πn-2π$

 

$∠C=\cos^{-1}{\left(\frac{{-{c}}^2+{a}^2+{b}^2}{2ab}\right)}$

 

$n=q\left(x\right)=\cos^{-1}{\left(\frac{{-{c}}^2+{a}^2+{b}^2}{2ab}\right)\ \cap\ f\left(x\right)=\cos^{-1}{\left(\frac{\left({-\sqrt[{x}]{{a}^x+{b}^x}}^2+{a}^2+{b}^2\right)}{2ab}\right)}}$

 

To illustrate, I have created a Geogebra animation found here: Non-Right Triangles Subjected to Pythagorean Generalization