For any triangle A1 B1 C1 where c* is the long leg, an exponent n exists that will satisfy a n + b n = c n of non-Diophantine concern. Vertex trisection of A1 B1 C1 through the proportional center yields areas where B1 C1 D + A1 C1 D = A1 B1 D. Subsequent dilation about the proportional center produces dissimilar quadrilaterals. Ad infinitum, these trapezia can exhibit corresponding conservation of the Pythagorean (et al.) Theorem if for a right triangle and the analogous for those absent perpendicularity.
*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 ∠C1 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.
Maintaining the Pythagorean area analogy as the interdependent n and ∠C1 deviate from the orthogonal, necessitates a departure from the conventional expression of area. Used here, rhombic or non-orthogonal area as defined by .5ab and given in units n, maintains harmony with the area of a right triangle while offering distinction from ½ base × height as well Heron.
Thus, the following assertion can be made regarding the requisite trapezia area of “Pythagorean” conformance:
**Shape is square if n = 2
Validity for this statement is found where the a : b ratio is expressed as the compliance exponent n and ∠C1 from:
Via a curve of function:
Further, while knowing the proportional center lies along a segment between the mid-points of a and b, I have yet to define it for other than right triangles and therefore use lateral trapezia height to establish the dilated triangle size and proportional center.
