Concerning the Harmonious Pythagorean Tetrahedra conjecture, please reference the prior post: PYTHAGOREAN THEOREM GENERALIZATION as CONSEQUENCE of THREE-DIMENSIONS
For any base right triangle of a:b ratio equal to or less than ¼:1, its area summed with that figured for each of its three sides compliant with the Pythagorean Theorem, is insufficient for three-dimensional existence.
Any base right triangle exceeding an a:b ratio of 1:1 represents a reciprocal dilation in the form b:a. Thus, a ¼:1 ratio serves as the two-dimensional threshold or limit while a 1:1 ratio can be examined as a constructive constraint with the region between the two containing all viable variation.
With circumspect confidence, my designate of proportional center, labeled point E, may hold promise as contribution to a proof. All point E’s lie in the locus of radius ¼:1, centered the same from the right corner along the side b.
