Is it true that for any given closed Jordan curve of $C \subset \mathbb{R}^2$ there is a dense subset $A$ such that for every point $p\in A$ we have the following property:


If we rotate $C$ around $p$ by 0 < $\alpha$ < $\pi$ clockwise on the plane, then the created curve $C'$ intersects $C$ at points other than $p$.


What if the angle of rotation becomes $\pi$ ?



Is there a dense subset on closed Jordan curve $C$ which its points make intersections under certain rotations?
Thanks for the efforts have done on this problem specially by @erz ,but It seems that *Mark J.Nielsen* have solved this problem [here](https://link.springer.com/article/10.1007%2FBF00151519) (that I have found it recently), while he was proving this theorem about inscribed triangles in closed simple jordan curves:


**[Theorem E](http://www.webpages.uidaho.edu/~markn/squares/):** *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.


