  Lower bounds estimates are proved for the first eigenvalue for the Dirichlet
Laplacian on arbitrary triangles using various symmetrization techniques. These
results can viewed as a generalization of P\'olya's isoperimetric bounds. It is
also shown that amongst triangles, the equilateral triangle minimizes the
spectral gap and (under additional assumption) the ratio of the first two
eigenvalues. This last result resembles the Payne-P\'olya-Weinberger conjecture
proved by Ashbaugh and Benguria.
