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

Global and Local Bounds on the Fundamental Ratio of Triangles and Quadrilaterals