Trouble With The Inversion Transformation

[Shoney45]

Homework Statement

1) Given a circle, construct another circle which is orthogonal to it. List the steps taken in this construction.

2) For a fixed circle and a point P not on the circle nor equal to the center of the circle, construct the image of P under inversion with respect to the circle. [You may wish to divide this problem into two cases, when P is exterior to the circle and when it is interior to the circle.]

3) Using 2), find the image of a line under inversion which is tangent to the circle of inversion.

4) State your finding as a theorem and prove it.

5) Using 2), find the image of a right triangle under inversion (consider the sides of the triangle as lines and not line segments). There are three cases to consider:
a. The center of inversion is a vertex of the triangle,
b. The center of inversion is on a side, but not a vertex, of the triangle, and
c. No side of the triangle contains the center of inversion.

Homework Equations

The Attempt at a Solution

So far I have completed numbers one and two. On number three, I have a couple of ideas, but I can't quite figure out how to map a point to its image under the transformation of inversion. I am using a circle of radius one, centered at the origin. I have the following sample mapping: P = (3,4) \rightarrowP' = (3/25, 4/25). I can see that the distance from the origin to P is 5. But I can't piece it together how the mapping works.

I'll address numbers four and five if I need to when I begin to try to tackle them.

 

[tiny-tim]

Hi Shoney45!

3) Using 2), find the image of a line under inversion which is tangent to the circle of inversion.

I'm guessing that it's going to be a circle …

if it is, then you know that it goes through the tangent point, and also that it goes through the inverse of infinity.