The Scaling Theorem A similarity scales all distances by a fixed positive factor.
Proof
Suppose first that C is not collinear with A and B.
Now suppose that D is on the line AB. Thus s scales all segments from A by the factor k = A'B'/AB. Similarly, s scales all segments from B by this factor. Since A and B were arbitrary, s scales all distances by the same factor.


Corollary Each similarity can be written as the composite of an isometry and a dilation about O.
Proof 
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