  We study the SL(2,R)-infimal lengths of simple closed curves on
half-translation surfaces. Our main result is a characterization of Veech
surfaces in terms of these lengths. We also revisit the "no small virtual
triangles" theorem of Smillie and Weiss and establish the following dichotomy:
the virtual triangle area spectrum of a half-translation surface either has a
gap above zero or is dense in a neighborhood of zero. These results make use of
the auxiliary polygon associated to a curve on a half-translation surface, as
introduced by Tang and Webb.
