Tiling a triangle into a prime number of congruent triangles

We prove that if a triangle is cut into $N$ congruent triangles and $N$ is prime, then $N=2$, $N=3$, or $N\equiv 1 \pmod 4$, and every such $N$ occurs. In particular, if $N > 3$ then $N$ is not a prime congruent to $3$ modulo $4$. The argument divides into cases according to possible shapes of the