  We prove that any sufficiently small perturbation of an isosceles triangle
has a periodic billiard path. Our proof involves the analysis of certain
infinite families of Fourier series that arise in connection with triangular
billiards, and reveals some self-similarity phenomena in irrational triangular
billiards. Our analysis illustrates the surprising fact that billiards on a
triangle near a Veech triangle is extremely complicated even though Billiards
on a Veech triangle is very well understood.
