  A computer-assisted proof is proposed for the Laplacian eigenvalue
minimization problems over triangular domains under diameter constraints. The
proof utilizes recently developed guaranteed computation methods for both
eigenvalues and eigenfunctions of differential operators. The paper also
provides an elementary and concise proof of the Hadamard shape derivative,
which helps to validate the monotonicity of eigenvalue with respect to shape
parameters. Besides the model homogeneous Dirichlet eigenvalue problem, the
eigenvalue problem associated with a non-homogeneous Neumann boundary
condition, which is related to the Crouzeix--Raviart interpolation error
constant, is considered. The computer-assisted proof tells that among the
triangles with the unit diameter, the equilateral triangle minimizes the first
eigenvalue for each concerned eigenvalue problem.
