Why does an isometry map any circle to a circle We took care

Why does an isometry map any circle to a circle? We took care to write the inverse of the isometry r_1r_2r_3 as r_3r_2r_1 because only this ordering of terms will always give the correct result.

Solution

An isometry T from R² to itself is a map that preserves distances between points. In other words ,

                                                d (Tp,Tq) = T(p,q).........................(1)

Now the circle C with center a and radius r is the set of points b such that

                                                 d(a,b) =r.......................................(2)

Applying (1) , we have             d(Ta,Tb) = d(a,b) =r.

That is , the image of the circle C is another circle with center Ta and same radius.

Thus any isometry maps circles to circles (in fact of the same radius)