Why Does the Pythagorean Theorem Work? | Steve Wait – January 25, 2021

Why does the Pythagorean Theorem work? Or perhaps a more pointed query… why only for the right triangle? Regrettably, and not surprisingly, I cannot advise conclusively. Nonetheless, I offer a vantage that may afford thought without mere recital of proofs. Regarding all triangles, the Law of Cosines notwithstanding, a more analogous with Pythagoras approach may be an exponent solution.

For any triangle ABC where c* is the long leg, an exponent n exists that will satisfy ${a}^n+{b}^n={c}^n$ of non-Diophantine concern.

*Alternatively, ${a}^n+{b}^n={c}^n$ can be re-written ${b}^n-{a}^n={c}^n$ at the isosceles inversion point (where leg c becomes equal to leg b) to maintain constancy of convention. This is necessitated as ∠C changes between the degenerate triangle limits of zero and π radians, the transition from obtuse scalene to acute scalene. Constrained by this equation, as triangles of a:b < 1 approach isosceles, n becomes extreme. In the case of a:b = 1, the equation is invalid at the equilateral where c fails to be the long leg.

If n is solved for $\sqrt{\left({a}^2+{b}^2-\left(2ab\ \times\ \cos{C}\right)\right)}=\sqrt[{n}]{{a}^n+{b}^n}$ and plotted against C, a curve of function $C=\cos^{-1}{\left(\frac{\left({-\sqrt[{n}]{{a}^n+{b}^n}}^2+{a}^2+{b}^2\right)}{2ab}\right)}$ emerges as an expression of a:b . If all ratios are represented, results intersect at C = 90° (or $\frac{\pi}{2}$) and n = 2; a kind of “pivot” point that all ratios rotate about. This coalescence, without consequence of leg ratio, distinctly illustrates Pythagoras’ (et al.) stature as being immune to the a:b ratio and held accountable only to that of the right angle. If agnostic ratios were to occur for $\sqrt[{n}]{{a}^n+{b}^n}$ solutions of a given C, the theorem’s greatness would be diminished.

I’ve created a Geogebra animation to better illustrate this construct. It can be viewed here: Why does the Pythagorean Theorem work? – Geogebra . The animation includes both the function, as described above, and the triangle ABC. It allows the user to manipulate the a:b ratio, thus depicting the changing functions rotation about C = 90° (or $\frac{\pi}{2}$) and n = 2. As well, the effects of this manipulation are seen on the graphed triangle ABC. The user may also move a point P along the function to show the changes in triangle ABC for a given a:b ratio. Note that in the animation the axes are swapped from the original manuscript graph and angle C is now shown in radians.