Permutation Generated Triangles: New Invariants of Squared Side Lengths and Additive Areas

Given three distinct real numbers p, q, r , all six permutations (a, b, c) of (p, q, r) are used to construct triangles with vertices (a, 1), (b, 2) , (c, 3) . The area admits the compact formula A=1/2[2b-(a+b)] ; consequently, the six permutations yield at most three distinct area values, each determined by the element occupying the middle position. Two novel contributions are established. First, the sum of the squares of the three side lengths remains constant across all six permutations, its value being expressible as a symmetric quadratic form in the three numbers plus a fixed constant. This invariance holds even though the triangles are generally not congruent. Second, the three distinct area values satisfy an additive relation: the largest area equals the sum of the two smaller ones. This property persists for signed areas when orientation is taken into account. The sum-of-squares invariant is generalized to arbitrary numbers of points and arbitrary ordinates, while the additive area relation is shown to depend on equal spacing of the ordinates. Rigorous algebraic proofs, numerical examples, geometric illustrations, and a concrete application to data integrity verification are provided.