  We prove $L^p$ norm convergence for (appropriate truncations of) the Fourier
series arising from the Dirichlet Laplacian eigenfunctions on three types of
triangular domains in $\mathbb{R}^2$: (i) the 45-90-45 triangle, (ii) the
equilateral triangle and (iii) the hemiequilateral triangle (i.e. half an
equilateral triangle cut along its height). The limitations of our argument to
these three types are discussed in light of Lam\'e's Theorem.
