In this talk I will explain how to construct counterexamples for two problems in spectral geometry. In the first part of the talk, I will explain how to prove that a triangle is not determined by its first, second and fourth (Dirichlet) eigenvalues, solving a conjecture by Antunes and Freitas. In the second part I will construct a planar domain with 6 holes for which the nodal line is closed and does not touch the boundary. In particular, this domain does not satisfy Payne’s nodal line conjecture. Both proofs are computer-assisted. Joint work with Joel Dahne, Kimberly Hou and Gerard Orriols.