# Introduction to Circle Inversions

## Explore the "Reflection about a circle tool" in geogebra

Plot the unit circle, centered at the origin, and label the center of the circle O. We will using the "reflection about the circle tool", which you will find by clicking "more" twice. It looks like this:

We will call the reflection about the circle O (also known as the

*inversion*about the circle O), .At what ordered pair is

At what ordered pair is ?

At what ordered pair is ?

Explore the images of other points, and describe any patterns you notice in how maps generic points. Inductive reasoning!

Now, make adjustments to the location of the center and the radius of the circle. Explore the image under the circle inversion of more points. What do you notice is *invariant *under the circle inversion?

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Suppose a circle with center has a radius of . Let be a point in the Euclidean plane that is distinct from . Define to be the point on ray that is distance from Verify that this definition for agrees with the reflect about circle tool for a point within the circle, a point outside the circle, and a point on the circle.

Describe how you verified the definition for agrees with the output of the reflect about the circle tool.

Is an isometry? How do you know?

Verify that the image of a line intersecting the center of a circle of inversion is itself.

Are the points on the line fixed? Explain.

Verify that the image of a line that does not intersect the circle of inversion is another circle that does intersect the circle of inversion.

What relationships do you notice about the line and its image under the inversion?

Verify the following statement using the reflection about a circle tool:
Triangle , with at the center of the circle, on the circle, and another point
is similar to Triangle

Prove this statement about the similarity is true.