  We continue the study of the analogue of Thurston's metric on the
Teichm{\"u}ller space of Euclidean triangle which was started by Saglam and
Papadopoulos in [1].By direct calculation, we give explicit expressions of the
distance function and the Finsler structure of the metric restricted to the
subspace of acute triangles.We deduce from the form of the Finsler unit sphere
a result on the infinitesimal rigidity of the metric.We give a description of
the maximal stretching loci for a family of extreme Lipschitz maps.
