An Inversion Construction that Works for all Points P

[It will be seen later, that this is based on the idea of Stereographic Projection.]

This construction is a good one for Sketchpad, for it works whether P is inside or outside the circle.

• This figure is constructed given the circle and the point P.
• Construct line OP and then construct the line through O perpendicular to OP, intersecting the circle in N and S. Then let E be the intersection of line NP with the circle (the intersection point distinct from N) and let Q be the intersection of lines OP and ES.
• The radius of the circle = r = |ON| = |OS|. To show Q is the inversion of P, one must show |OP||OQ| = r^2. This can be proved from the similar triangles PON and SOQ. To show the triangles are similar, show that each is similar to PEQ.