Hi Mac,

One simple way to approach this is to group the edges by length, and shrink/expand each edge towards the average of its group. Clearly a fixed number of edge lengths leads to a fixed number of triangles.

Here's a mesh before this clustering:

and after:

In general this won't lead to a very smooth mesh, but you can get strict edge lengths.

I attached the definition for this below. There is a slider for how strongly vertices try and keep near their original locations - by adjusting this you can vary the priority of exact edge lengths vs closeness to the original shape.

This is just a start though - there's lots more work that could be done along this direction.

For instance, instead of fixing the target lengths once at the start, the groupings could be adjusted iteratively using something like K-Means clustering.

Slightly more involved would be to make a custom goal which adjusts whole triangles instead of edges.

It is sometimes helpful to think of triangle shapes as points in a three dimensional space, where the coordinates correspond to edge lengths. However, you need to also account for the fact that a 3,4,5 triangle is the same as a 4,5,3 triangle, and so on.