Given: circle x with a radius r and circle y with radius s prove: circle x is similar to circle y
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Puzzling questions actually! I always considered the fact that all circles are similar by definition. Wanting to give it a rationale tho, if two shapes are similar, the lenght of one is scaled by the sides ratio, and their surface is scaled by the square of the ratio(easiest to see with triangles, if you pick two triangles, where the second's lenght are twice the first, second's perimeter is twice the first, and the area is four times the first); and it's true the opposite: if two shapes have their border and surface with said ratios, the two are similar.

Now, be K the ratio between the radii, or .

The lenght of the first circomference will be , and the lenght of the second The ratio between the two will be

The area of the first circle will be , while the area of the second is . Their ratio is, as before,

That might prove that the two are similar.