  Due to the invariance properties of cross-ratio, M\"obius transformations are
often used to map a set of points or other geometric object into a symmetric
position to simplify a problem studied. However, when the points are mapped
under a M\"obius transformation, the distortion of the Euclidean geometry is
rarely considered. Here, we identify several cases where the distortion caused
by this symmetrization can be measured in terms of the Lipschitz constant of
the M\"obius transformation in the Euclidean or the chordal metric.
