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 with absolute conclusion. 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 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 √(a 2 + b 2 – ( 2ab x cos C )) = ( a n + b n ) (1/n) and plotted against C, a curve of function C = cos-1 ( ( -( ( ( a n + b n ) (1/n) ) 2 ) + a 2 + b 2 ) / 2ab ) emerges as an expression of a : b . If all ratios are represented, results intersect at C = 90°, n = 2. This coalescence, without consequence of leg ratio, distinctly illustrates Pythagoras’ (et al.) stature. If agnostic ratios were to occur for ( a n + b n) (1/n) solutions of a given C, greatness would be diminished.
The Geogebra animation can be viewed here: Why does the Pythagorean Theorem work? – GeoGebra Note that in the animation (screen shot below) the axes are swapped from the original graph and angle C is given in radians.
Additionally, yet not fully explored, by the same method discussed in prior posting, three-dimensional solutions (tetrahedra in the form of oblique triangular pyramids) can be found for non-right triangles bases.
Please reference the prior post concerning the Harmonious Pythagorean Tetrahedra conjecture: PYTHAGOREAN THEOREM GENERALIZATION as CONSEQUENCE of THREE-DIMENSIONS
