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Thanks for the efforts have done on this problem specially by @erz ,but It seems that Mark J.Nielsen have solved this problem here (that I have found it recently), while he was proving 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 Here.

Thanks for the efforts have done on this problem specially by @erz ,but It seems that Mark J.Nielsen have solved this problem here (that I have found it recently), while he was proving 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.

Actually the statement of this theroem is equal to the problem above ,therefore the answer of this problem is affirmative.

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.

Thanks for the effort have done on this problem specially by @erz ,but It seems that Mark J.Nielsen have solved this problem here (that I have found it recently), while he was proving 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 Here.

Thanks for the efforts have done on this problem specially by @erz ,but It seems that Mark J.Nielsen have solved this problem here (that I have found it recently), while he was proving 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 Here.

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 Here.

Thanks for the effort have done on this problem specially by @erz ,but It seems that Mark J.Nielsen have solved this problem here (that I have found it recently), while he was proving 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 Here.