Part A: A circle is the set of all points that are the same distance from one given point. Find an example that contradicts this definition. How would you change the definition to make it more accurate?

We have the definition:

"A circle is the set of all points that are at the same distance from one given point"

This, in fact, can define a circle, but can also define other shapes, because we are not defining the dimension.

So, for example, if we are in 3D, then the above definition does not define a circle, it will define a sphere.

Then we should also add the restriction that we should be in 2-dimensions.

So we could say:

"A circle is the set of all points in a given plane that are at the same distance from one given point on that plane"

Now, because all the points are in the same plane, we know that we are un 2D, so this will always define a circle.