**Geometric Inversion: A Conceptual Foundation**

Geometric inversion is means of a transforming points in a plane with respect
to an arbitrary circle, called the circle of inversion. For the purposes of
this unit on geometric inversion, the circle of inversion shall be called **gamma** and
have center **O **and radius **r**. Geometric inversion
transforms each point **P** in the plane with into a point **P'**,
also in the plane, such that **O**, **P**, and **P'** are collinear and the
following
relationship is true:

(**OP**)(**OP'**) = **r**^{2}

The figure below depicts **P** and **P'** with
respect to gamma.

To download a Geometer's Sketchpad sketch with this figure,
**Click Here**. Using this sketch, explore the model by changing the position
of **P**.

1. What ifPis located in the interior of gamma?

2. What ifPis on gamma?

3.Where is the inverse point ofO?

4. Can you verify the conjectures you made in questions 1, 2, and 3 using the relationship betweenOP,OP', andr?

PandP'will maintain a relationship where one is partitioned outside gamma and one is inside. WhenPis in the interior of gamma,OP<r. ThusOP'>rfor the relationship between the three distances to hold. WhenPis on gamma,OP=r. Thus,OP'=ras well.Ohas no inverse point. IfPwere located atO, thenOP= 0. This quantity, when substituted in the relationship causesOP'to be undefined. A good way to discuss this case is to movePclose toOand examine the change inOPandOP'.

To continue with the introduction by examining the analytic geometry of geometric
inversion, **Click Here**.

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