# 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 $f(P)$ of the plane into itself such that $d(f(P),f(Q))=d(P,Q)$.

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

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

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

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 $O$. So we know that $d(O,x)=r$ for every $x\in C$ (our circle) and so $d\left(f(x),f(O)\right)=d(x,O)=r$ for every $x\in f(C)$.
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!