  We study equivalence relation of the set of triangles generated by similarity
and operation on a triangle to get a new one by joining division points of
three edges with the same ratio. Using the moduli space of similarity classes
of triangles introduced by Nakamura and Oguiso, we give characterization of
equivalent triangles in terms of circles of Apollonius (or hyperbolic pencil of
circles) and properties of special equivalent triangles. We also study
rationality problem and constructibility problem.
