In this project we will investigate how to invert a point
through a circle. The goal of this project is for students to gain facility
with the idea of circle inversion.
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 at the top of the window. 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?