arXiv:2611.14803v1 Announce Type: new Abstract: A $\Delta$-triple for a natural number $n$ is a triple $(x,y,z)$ of divisors of $n$ with $1<x<y\leq z<\sqrt n$ and $n/x-x=(n/y-y)+(n/z-z)$. The numbers that admit such a triple are those with the $\Delta$ property, which governs the existence of certain $n$-simple triangles in factor graphs of split graphs. We allow the three components to be rational numbers, and show that, once the quotient $z/y=a/b$ is fixed, the rational $\Delta$-triples for $n$ are exactly the rational points of the elliptic curve $Y^2=n\,ab\,w(w+ab)\bigl(w+(a+b)^2\bigr)$ lying over the interval $-b(a+b)<w<-ab$. Since a rational $\Delta$-triple for $n$ amounts to $\Delta$-triples for infinitely many of the numbers $\kappa^2n$, the chord--tangent construction becomes a way of producing new $\Delta$-triples from old ones. For perfect squares, the curve always carries a rational point coming from a degenerate triple. This point has order $4$ if $a=b$, and infinite order if $a>b$. As a consequence, for every quotient $a/b>1$ there are infinitely many pairwise non-proportional $\Delta$-triples of perfect squares with $z/y=a/b$, in contrast with the fact that perfect squares have no $\Delta$-triple with $y=z$. For arbitrary $n$, there are infinitely many rational $\Delta$-triples for $n$ with a given quotient if and only if the corresponding curve has positive rank.
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