# Thread: Prove that every rigid motion transforms circles into circles

1. ## Prove that every rigid motion transforms circles into circles

2. Originally Posted by Pinkk
What's your definition of rigid motion?

3. A rigid motion of the Euclidean plane is a transformation $\displaystyle f(P)$ of the plane into itself such that $\displaystyle d(f(P),f(Q))=d(P,Q)$.

Would I just write that if you are given a circle $\displaystyle O$ with center $\displaystyle A$ and any arbitrary point on the circle $\displaystyle B$ that is always distance $\displaystyle \alpha$ from the center, then according to the definition of rigid motion, $\displaystyle d(f(A),f(B))=d(A,B)=\alpha$, so any arbitrary point $\displaystyle B'$ on $\displaystyle O'$ is distance $\displaystyle \alpha$ from $\displaystyle A'$, and so $\displaystyle O'$ is a circle?

4. Originally Posted by Pinkk
A rigid motion of the Euclidean plane is a transformation $\displaystyle f(P)$ of the plane into itself such that $\displaystyle d(f(P),f(Q))=d(P,Q)$.

Would I just write that if you are given a circle $\displaystyle O$ with center $\displaystyle A$ and any arbitrary point on the circle $\displaystyle B$ that is always distance $\displaystyle \alpha$ from the center, then according to the definition of rigid motion, $\displaystyle d(f(A),f(B))=d(A,B)=\alpha$, so any arbitrary point $\displaystyle B'$ on $\displaystyle O'$ is distance $\displaystyle \alpha$ from $\displaystyle A'$, and so $\displaystyle O'$ is a circle?
Oh...and isometry . A set of points forms a circle if all the points are equidistant from a signle point $\displaystyle O$. So we know that $\displaystyle d(O,x)=r$ for every $\displaystyle x\in C$ (our circle) and so $\displaystyle d\left(f(x),f(O)\right)=d(x,O)=r$ for every $\displaystyle x\in f(C)$.

5. Originally Posted by Drexel28
Oh...and isometry . A set of points forms a circle if all the points are equidistant from a signle point $\displaystyle O$. So we know that $\displaystyle d(O,x)=r$ for every $\displaystyle x\in C$ (our circle) and so $\displaystyle d\left(f(x),f(O)\right)=d(x,O)=r$ for every $\displaystyle x\in f(C)$.
Ah okay. I had a vague idea that that must have been the approach, but I was having a problem formulating it into an actual proof. Thanks!