Editing Inversive geometry (section)

==== Compass and straightedge construction ====
[[File:Inversion in circle.svg|thumb|To construct the inverse ''P{{'}}'' of a point ''P'' outside a circle ''Ø'':  Let ''r'' be the radius of ''Ø''. Right triangles ''OPN'' and ''ONP{{'}}'' are similar. ''OP'' is to ''r'' as ''r'' is to ''OP{{'}}''.]]

===== Point outside circle =====
To [[Compass and straightedge constructions|construct]] the inverse ''P{{'}}'' of a point ''P'' outside a circle ''Ø'':
* Draw the segment from ''O'' (center of circle ''Ø'') to ''P''.
* Let ''M'' be the midpoint of ''OP''. (Not shown)
* Draw the circle ''c'' with center ''M'' going through ''P''. (Not labeled. It's the blue circle)
* Let ''N'' and ''N{{'}}'' be the points where ''Ø'' and ''c'' intersect.
* Draw segment ''NN{{'}}''.
* ''P{{'}}'' is where ''OP'' and ''NN{{'}}'' intersect.

===== Point inside circle =====
To construct the inverse ''P'' of a point ''P{{'}}'' inside a circle ''Ø'':
* Draw ray ''r'' from ''O'' (center of circle ''Ø'') through ''P{{'}}''. (Not labeled, it's the horizontal line)
* Draw line ''s'' through ''P{{'}}'' perpendicular to ''r''. (Not labeled. It's the vertical line)
* Let ''N'' be one of the points where ''Ø'' and ''s'' intersect.
* Draw the segment ''ON''.
* Draw line ''t'' through ''N'' perpendicular to ''ON''.
*  ''P'' is where ray ''r'' and line ''t'' intersect.