A Third Pell Equation and Elliptic Restrictions in the Family of Consecutive Heronian Triangles

This paper introduces and analyzes a third Pell-type equation naturally arising in the geometry of consecutive Heronian triangles (a−1,a,a+1)(a-1,a,a+1)(a−1,a,a+1).
Starting from the classical area condition

k^2 - 3a^2 = -12

and the secondary decomposition relation

m^2 - 3u^2 = 6

we show that the global inradius rrr satisfies a third independent Pell-type conic:

6r^2 - 6ru + u^2 = 1

We prove that these three Pell equations are fully equivalent via explicit linear bijections connecting the variables (a,k)(a,k)(a,k), (u,m)(u,m)(u,m), and (u,r)(u,r)(u,r).
Next, we impose additional arithmetic restrictions—such as requiring the inradius, an exradius, or certain geometric products to be perfect squares—and show that each produces an elliptic curve with finitely many integer points.

Analyzing these corresponding elliptic curves reveals that the unique nondegenerate solution within the entire family of consecutive Heronian triangles is the triangle 13–14–15. This suggests a deep connection between Pellian structures, elliptic curves, and discrete triangle geometry.

We provide a detailed outline of the elliptic curve framework and invite specialists in Diophantine geometry to collaborate toward a full classification.