  We present a new, computer-assisted, proof that for all triangles in the
plane, the equilateral triangle uniquely maximizes the ratio of the first two
Dirichlet-Laplacian eigenvalues. This proves an independent proof the
triangular Ashbaugh-Benguria-Payne-Polya-Weinberger conjecture first proved in
arXiv:0707.3631 [math.SP] and arXiv:2009.00927 [math.SP]. Inspired by
arXiv:1109.4117 [math.SP], the primary method is to use a perturbative estimate
to determine a local optimum, and to then use a continuity estimate for the
fundamental ratio to perform a rigorous computational search of parameter
space. We repeat a portion of this proof to show that the square is a strict
local optimizer of the fundamental ratio among quadrilaterals in the plane
