1.  In the window below we have constructed a circle C with center O and radius point R.  Note that this is a "live" Geometry Explorer Java applet window, so you can grab points and move them around to see how the relationships among the figures change.

2.  Point P is placed inside the circle, followed by the construction of ray OP.   Next, at P we construct a perpendicular to ray OP, and find the intersection points (T and U) of this perpendicular with the circle.  We connect O and T with a segment and construct a perpendicular to OT at T.  Finally, we let P' be the intersection point of this perpendicular with ray OP.

3.  Note the measurements in the upper left-hand corner.  In particular note that the product of OP and OP' is exactly the same as the radius of the circle squared.  Drag point P and/or radius point R around and note that this special relationship among OP, OP' and the radius persists.  We call point P' the Inverse of P with respect to the circle C if (OP)*(OP') = (radius of C)^2.

4. What will happen as point P approaches the center of the circle?  (Do you see where the title of this project comes from?)  What will happen as point P approaches the boundary of the circle?