  Given two triangles whose angles are all acute, we find a homeomorphism with
the smallest Lipschitz constant between them and we give a formula for the
Lipschitz constant of this map. We show that on the set of pairs of acute
triangles with fixed area, the function which assigns the logarithm of the
smallest Lipschitz constant of Lipschitz maps between them is a symmetric
metric. We show that this metric is Finsler, we give a necessary and sufficient
condition for a path in this metric space to be geodesic and we determine the
isometry group of this metric space. This study is motivated by Thurston's
asymmetric metric on the Teichm{\"u}ller space of a hyperbolic surface, and the
results in this paper constitute an analysis of a basic Euclidean analogue of
Thurston's hyperbolic theory. Many interesting questions in the Euclidean
setting deserve further attention.
