It seems that Mark J.Nielsen have solved this problem here that I have found it recently, where he was proved this theorem about inscribed triangles in closed simple jordan curves:
Theorem E: Extending Theorem D, J has so many inscribed triangles similar to T that the vertices of all these inscribed triangles are "dense" in the curve J.
As the article needs license I do not have access to the whole solution ,I think it would be great to present and discuss about the way the problem has been solved.