What makes circles similar

Take any two circles, and slap some Cartesian Coordinates on them, such that the first is at the origin. Translate the second circle to the origin, then dilate it until the radii match.

Thus the pair of circles is similar. That explanation sounds like circular reasoning to me. How is that a proof? You state what similar means and then say since they are the same shape, they are similar. Are there any axioms involved? The definition is based on transformations and congruence. The axioms are not being invoked directly, but everything done is undergirded by them. Traditionally, Euclidean Geometry did not depend on a coordinate system for meaning. This is new to me.

When, and who, made this decision? I gave my answer. In no way did I suggest it is the only proof. One of the beautiful things about math is that there are sometimes many ways to prove the same thing. So no, Euclidean geometry is not dependent on Cartesian coordinates, but coordinate proofs are one of the tools in the toolbox even of the high school geometry student.

Additionally, I was outlining , rather than detailing the proof, so it may have looked like handwaving because I was allowing the reader to fill in the details. Since it seems more clarity is called for, feel free to see my extended explanation below: To show: All circles are similar. Since a combination of two sequences of the above transformations is still a sequence of the above transformations, this would succeed in showing that the two circles are similar.

Without Loss Of Generality place a circle in the plane centered at h,k with radius r. This version has less handwaving. It also has a lot more notation and makes the concept of the proof ugly and obscured. I think the expectation for a high school student is more along the lines of my original argument:. You can take two circles and move one so that they have the same center, then dilate it so they are the same size.

Why must such a simple idea be made so complicated? Might such a definition also apply to circles? I might try using the same reasoning I did for the circles here. However, it's not obvious to me how finding the properties of similarity for ellipses helps here.

Add a comment. Active Oldest Votes. Arthur Arthur k 14 14 gold badges silver badges bronze badges. Given two circles are you able to find a bijection which transforms one circle into another?

Community Bot 1. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post. Related 0. Since similarity is independent of position and circles remain the same regardless of rotation or reflection , the similarity of circles depends only on the circle radii. But wait!

Since a radius is a constant an unchanging number , and any constant is proportional to another constant, then all circles must be similar. A related geometric property is congruence. Congruent shapes are identical in every possible way: proportion, angles, size.

Like similarity, congruence is unaffected by position, rotation, or reflection. Similarity of Circles - Expii All circles are similar to one another. Geometry Circle Concepts. Similarity of Circles. Go to Topic. Explanations 4. Related Lessons. Which Circles Are Congruent?