An interesting discussion was started on the geom.puzzles newsgroup with the following letter. For example, the center of a circle is an equichordal point for that circle. This is clear, but I can think of no other example. Thank you, anyone. As it came out, Mary's question was related to a problem that has been around for more than 80 years and was solved as recently as 1997 by Marek Rychlik from University of Arizona.
P has the same length.
Rychlik solved the problem for a more general class of regions. He showed that no star - shaped region may contain more than one equichordal point. This does not answer Mary's question, though. And there are many such curves. Imagine a fixed length rod constrained to rotate around a fixed point but let the rod be free to slide towards any of its ends.