  Despite the moduli space of triangles being three dimensional, we prove the
existence of two triangles which are not isometric to each other for which the
first, second and fourth Dirichlet eigenvalues coincide, establishing a
numerical observation from Antunes-Freitas [P. R. S. Antunes and P. Freitas.
Proc. R. Soc. Lond.Ser. A Math. Phys. Eng. Sci., 467(2130):1546-1562, 2011].
The two triangles are far from any known, explicit cases. To do so, we develop
new tools to rigorously enclose eigenvalues to a very high precision, as well
as their position in the spectrum. This result is also mentioned as (the
negative part of) Conjecture 6.46 in [R. Laugesen, B. Siudeja, Shape
optimization and spectral theory, 149-200. De Gruyter Open, Warsaw, 2017], Open
Problem 1 in [D. Grieser, S. Maronna, Notices Amer. Math. Soc.,
60(11):1440-1447, 2013] and Conjecture 3 in [Z. Lu, J. Rowlett. Amer. Math.
Monthly, 122(9):815-835, 2015.].
