Question

Translate one circle so that its center coincides with the other's, then perform a dilation (homothety) about that common center with ratio r2/r1 where r1,r2 are the radii; this maps the first circle exactly onto the second and thus every pair of circles are related by a similarity transformation and hence are similar.

Answer 1

Let C1 have center O1 and radius r1, and C2 have center O2 and radius r2 (r1,r2>0). Define the map
f(x) = O2 + (r2/r1) (x − O1).

This is the composition of the translation x ↦ x + (O2 − O1) (bringing O1 to O2) followed by the dilation (homothety) about O2 with factor r2/r1. For any point X on C1,
|f(X) − O2| = (r2/r1) |X − O1| = (r2/r1) r1 = r2,
so f maps C1 exactly onto C2. Since f is a similarity (translation + dilation), the two circles are similar.