Suppose there are two triangles on a plane. The coordinates for each point of both triangles are the same.
It seems to me that there is nothing to differentiate these two triangles, and as such, they must be identical. Therefore, there can't be more than one such triangle on the plane.
I might describe two congruent triangles positioned such that either can rotate to overlap the other one perfectly. In that case, it seems that I could say that the two triangles are identical after the rotation.
I'm in first year, so I can only speculate about how set theory represents geometry, but I have a hunch that something would disappear from a set after the rotation, which would decrease the cardinality of the set. However, if there were only one triangle, I suspect that you could rotate it all you want without affecting the cardinality of a set. It would seem very weird if rotation could end the existence of a triangle in some circumstances, but not in others. Then again, perhaps that's just a consequence of the rules.
Nevertheless, I can imagine an instructor asking a student to 'rotate the triangle so that it exactly overlaps the other one, record the angle of rotation, dilate it by a factor of x, and then sketch the result. In this case, the instructor seems to talk about the plane as though it includes two identical triangles, even after the rotation; it also seems that the correct sketch would include two triangles, one inside the other.
Can a plane include two indiscernible triangles?