  For each right-angled hexagon in the hyperbolic plane, we construct a
one-parameter family of right-angled hexagons with a Lipschitz map between any
two elements in this family, realizing the smallest Lipschitz constant in the
homotopy class of this map relative to the boundary. As a consequence of this
construction, we exhibit new geodesics for the arc metric on the
Teichm{\"u}ller space of an arbitrary surface of negative Euler characteristic
with nonempty boundary. We also obtain new geodesics for Thurston's metric on
Teichm{\"u}ller spaces of hyperbolic surfaces without boundary.
