I am looking for a minimal number of properties describing a triangle so that these properties are invariant to the choice of a Cartesian coordinate system as well as to the order in which the triangle points are enumerated. I have a few thousands of data points each of which is a triplet of multidimensional vectors. I thought to treat them as triangles. Not all of them are different. Some are either rotated or flipped versions of others. I wanted to distinguish them by their invariants.

I thought about using distances between certain triangle centers such as the center of the incircle, the circumcenter, the orthocenter, the centroid, etc. However, I found that the number of such centers is very large (around 40,000 listed in the Encyclopedia of triangle centers at the date I am writing this question).

Is there a small subset of those triangle centers, distances between which would unambiguously distinguish one triangle from another? What is the minimal amount of such triangle centers? Are there any papers that have information about them?