  For an elliptic curve $E$ defined over the field $\mathbb{C}$ of complex
numbers, we classify all translates of elliptic curves in $E^3$ such that the
$x$-coordinates satisfy a linear equation. This classification enables us to
establish a relation between the rank of finite rank subgroups of $E$ and
triples in $E$ whose $x$-coordinates are linearly related. The method of proof
integrates complex analytic techniques on elliptic curves with results of Gao,
Ge and K\"uhne on Uniform Mordell-Lang Conjecture for subvarieties in abelian
varieties.
