# Show that any two circles in the plane with the same center are Bertrand curves.

$$\alpha(s)$$ and $$\beta(s)$$ are called Bertrand curves if for each $$s_0$$, the normal line to $$\alpha$$ at $$s = s_0$$ is the same as the normal line to $$\beta(s)$$ at $$s = s_0$$. ($$s$$ need not be arc length on both $$\alpha$$ and $$\beta$$.) We say that $$\beta$$ is a Bertrand mate for $$\alpha$$ if $$\alpha$$ and $$\beta$$ are Bertrand curves.

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