**Inverses of Points**

Consider the case where the circle gamma is centered at the origin with radius
**r**. Let **P** and **P'** have the coordinates (**x**, **y**) and (**x'**, **y'**), respectively.

We
know that, since **P** and **P'** are inverse points
with respect to gamma, (**OP**)(**OP'**)
= **r**^{2}. Using the coordinates of **P** and **P'**,
this relationship can be reexpressed as:

1. Find (x',y'), the coordinates ofP', in terms ofxandy.

2. How would you modify the above equations and the coordinates you found in question 1 so that the center of gamma is at a point other than the origin?

Similar triangles can be formed by dropping perpendiculars fromPandP'to the x-axis, creating the proportion below.

Substituting the value ofx'^{ }^{2}from the equation above,

By factoring and simplifying, the value fory'is found with respect tox,y, andr. By a similar process,x'is also found:

Note that equations forxandywith respect tox',y', andrwill be similar in structure.

When gamma is not centered at the origin, let the center of gamma be (h,k). The equations forx'andy'are: