  We study the distribution of the height of the intersection between the
projective line defined by the linear polynomial $x_{0}+x_{1}+x_{2}$ and its
translate by a torsion point. We show that for a strict sequence of torsion
points, the corresponding heights converge to a real number that is a rational
multiple of a quotient of special values of the Riemann zeta function. We also
determine the range of these heights, characterize the extremal cases, and
study their limit for sequences of torsion points that are strict in proper
algebraic subgroups.
  In addition, we interpret our main result from the viewpoint of Arakelov
geometry, showing that for a strict sequence of torsion points the limit of the
corresponding heights coincides with an Arakelov height of the cycle of the
projective plane over the integers defined by the same linear polynomial. This
is a particular case of a conjectural asymptotic version of the arithmetic
B\'ezout theorem.
  Using the interplay between arithmetic and convex objects from the Arakelov
geometry of toric varieties, we show that this Arakelov height can be expressed
as the mean of a piecewise linear function on the amoeba of the projective
line, which in turn can be computed as the aforementioned real number.
