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.  In this figure we have constructed a point P outside the circle.  Also, we have constructed two rays from P that intersect the circle at the points P1, P2, Q1, and Q2.  Next, we have measured the distances listed in the upper left corner. Finally, we have calculated the product of P P1 and P P2  and also  the product of P Q1 to P Q2.   Move point P around.  Notice that these two products are always equal to one another.  Why should this be the case?  (Hint: Use the fact that triangles PP1Q2 and PQ1P2 are similar.)

3. Define the Power of P with respect to C to be the square of the distance from P to O minus the square of the radius of C.  It is not hard (a little algebra) to show that  (P P1)*(P P2) = (Power of P with respect to C) and thus the power of P is the same value as the two products we have in the window above.  Can you think of a way to use the power of P to classify points as being outside the circle vs on the circle?