Thanks for the effort have done on this problem specially by @erz  ,but It seems that *Mark J.Nielsen* have solved this problem [here][1] (that I have found it recently), while he was proving this theorem about inscribed triangles in closed simple jordan curves:

**[Theorem E][2]:** *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 Here.


  [1]: https://link.springer.com/article/10.1007%2FBF00151519
  [2]: http://www.webpages.uidaho.edu/~markn/squares/