At first, when thinking about similar circles I tried extending the properties of similar triangles to circles to reason that all circles are similar.
The properties of similar triangles are as follows:
1) Corresponding sides must be proportional. (That is, the ratios of their corresponding sides must be equal).
2) Corresponding angles must have the same measure.
Each side of a triangle must be scaled up or down by the same factor to get a "new" triangle that has proportional corresponding sides to the "original" triangle.
I figured a similar thing must happen between similar circles. If the radius of a circle is scaled up or down by some factor then its circumference will also be scaled up or down by that same factor. The "new" circle formed will be a scaled up or down version of the "original" circle, and its radius and circumference will be proportional to the "original's" radius and circumference. In fact, the ratios between their respective circumferences and radii would both simplify to $2\pi$.
Any two circles, regardless of size, will always have the same central angle of $360^\circ$.
Therefore, I concluded that all circles are similar by extending the properties of similar triangles.
Granted, this isn't a rigorous way of defining nor proving the similarity between all circles, but is my "intuition"/rough reasoning correct?
Also, what are the properties of similar circles? Are they at all akin to the properties of similar triangles?
Best Answer
If you want to define similarity of general shapes, not just polygons, you have to be quite rigorous.
The standard way is to simply list all possible transformations which intuitively don't affect similarity, and then define two things to be similar if they are connected by such a transformation.
In standard Euclidean geometry, these transformations are usually taken to be translations, uniform scalings, mirrorings, rotations, and any (finite) combination of such.
So any two circles are similar because you can translate and scale one to make it into the other.