- #1

Fredrik

Staff Emeritus

Science Advisor

Gold Member

- 10,851

- 413

My first thought is to define ##g=f-f(0)##, prove that g is a bijection that preserves distances too, and then try to prove that g is a rotation. I recently learned how to prove that maps that take straight lines to straight lines are linear, so I'm guessing that the strategy involves proving that g takes straight lines to straight lines.

Maybe we can use that g preserves circles around 0 in the sense that if C is such a circle and ##x\in C##, then ##g(x)\in C##. I imagine that we would take an arbitrary line L, and then look at points where it intersects some circle around 0, but I don't see how to turn this idea into a proof. I'm not sure if I should try to prove that g takes straight lines to straight lines first, or if there's a more direct approach.