TITLE: Can a circle be divided into congruent shapes with one not sharing a point with circumferences?
QUESTION [5 upvotes]: Just wondering out of curiosity. For instance, this does not qualify:

Having been trying for a while, I would appreciate if someone can give a proof that it is impossible (in case it is). Thanks in advance :).

REPLY [7 votes]: If you mean at least one rather than exactly one, then the second tiling below qualifies:

This is from Haddley and Worsley, "Infinite families of monohedral disk tilings" (2015), which I found via the similar MathOverflow question "Is it possible to dissect a disk into congruent pieces, so that a neighborhood of the origin is contained within a single piece?". That question seems to still be an open problem.