TITLE: Prove the theorem on analytic geometry in the picture.
QUESTION [104 upvotes]: I discovered this elegant theorem in my facebook feed. Does anyone have any idea how to prove?
Formulations of this theorem can be found in the answers and the comments. You are welcome to join in the discussion.
Edit:
Current progress: The theorem is proven. There is a bound to curvature to be satisfied before the theorem can hold.
Some unresolved issues and some food for thought:
(1) Formalise the definition (construction) of the parallel curve in the case of concave curves. Thereafter, consider whether this theorem is true under this definition, where the closed smooth curve is formed by convex and concave curves linked alternatively.
(2) Reaffirm the upper bond of $r$ proposed, i.e. $r=\rho$, where $\rho$ is the radius of the curvature, to avoid self-intersection.
(3) What is the least identity of the curves in order for this theorem to be true?
This is similar to the first part of question (1).
(4) Can this proof be more generalised? For example, what if this theorem is extended into higher dimensions? (Is there any analogy in higher dimensions?)
Also, I would like to bring your attention towards some of the newly posted answers.
Meanwhile, any alternative approach or proof is encouraged. Please do not hesitate in providing any insights or comments in further progressing this question.

REPLY [16 votes]: Consider:

The difference in length of $2{\pi}r$ is obvious because $2{\pi}(a+r)-2{\pi}{a}=2{\pi}r$.
Now say we pinch both the circles at the same time to the same extend, causing two points to travel the same displacement. If they travel the same displacement than the points separation of $r$ will still be there. We can pinch the circles many times, and by doing that we can end up with your shapes.