Representing triangle occurrence, the curves shown of ∠C on the graph cover a range from obtuse to the acute above the isosceles inversion. Regarding the scalene, the influence of both a:b ratio and ∠C is apparent when administered by an+bn=cn. Noted, the same will hold for the application of bn-an=cn to the acute remainder below the inversion. As a:b approaches 1:1, diminishing sensitivity to C becomes discernable. Contrastingly, the Pythagorean Theorem enjoys immunity to the a:b ratio and is held accountable only to the right angle. The degenerate locus curve delineates the regions of tetrahedral solutions (green) and that of non-occurrence (orange).
Please reference prior posts: WHY DOES the PYTHAGOREAN THEOREM WORK? and The ISOSCELES INVERSION POINT of an+bn=cn
Curves depicted for ∠C represent triangle solutions having conformity with (an+bn)(1/n) =√a2+b2-(2ab x cos C) or C=cos-1((-(((an+bn)(1/n))2)+a2+b2)/2ab ). Employed is an+bn=cn where c is the long leg for obtuse scalene triangles and some acute scalene triangles above the isosceles inversion, and bn-an=cn is used below the inversion point for the remainder of acute triangles. Tetrahedrons, being oblique triangular pyramids, may exist in the region of three-dimensional solutions shown. They may evolve from either right or scalene triangle bases and will possess corresponding lateral face areas consonant with a2+b2=c2 or an+bn=cn as applicable. The two-dimensional locus boundary (V=0) established by the distance of 2c being equal to the perpendicular distance from c to the intersection of lines constructed parallel to a and b at +2a and -2b respectively.
Many thanks to my colleague Mr. Sawyer Awald for his MATLAB® skills in determining the 2D degenerate locus.
