Comment by Gro-Tsen on Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions

This is reminiscent of a kind of “iterated mean” process (see this question on this subject), but in dimension 2 and invariant under Euclidean similitudes rather than in dimension 1. To prove convergence you probably need to rephrase it as a map in the modulus space of triangles, but it is unlikely that the point will have interesting properties, as already the arithmetic-quadratic mean of two real numbers is largely unstudied (as per previously linked question).