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Math Help - Prove that every rigid motion transforms circles into circles

  1. #1
    Senior Member Pinkk's Avatar
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    Prove that every rigid motion transforms circles into circles

    I do not know how to go about this. Any help would be appreciated.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pinkk View Post
    I do not know how to go about this. Any help would be appreciated.
    What's your definition of rigid motion?
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    Senior Member Pinkk's Avatar
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    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?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pinkk View Post
    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).
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    Senior Member Pinkk's Avatar
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    Quote Originally Posted by Drexel28 View Post
    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!
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