# 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.

## 1 Answer

A line is normal to a circle if and only if it passes through its center. As the two circles are supposed to have the same center, you're done taking for parameter the angle of the points on the circles.