There are 1, 4, 23, 180, 1806, 20198, ... (OEIS A056814) topologically distinct ways to divide a triangle into , 3, ... smaller triangles (Vicher).
A triangle partition is prime if it does not contain a triangle partition of lower order. The number of prime triangle partitions of orders , 3, ... are 1, 1, 3, 8, 62, 535, 4213, ... (OEIS A053740).
A specific type of triangle dissection consists of triangle together with an interior point such that the original side lengths and the additional three segments created by connecting the triangulation point with the vertices are all integers. An example of such a dissection is illustrated above (Pegg).
Other possible dissections allow cut lines to be drawn from arbitrary points along the sides. Allowing only primitive triangles without any parallel lines, isosceles triangles, or similar triangles, the smallest 3-piece integer dissection of each of the four possibly types is illustrated above.
Similarly, the smallest 4-piece dissection for each of the 23 possible 4-piece dissections is shown above.
Two 5-piece dissections are illustrated above (Pegg).