Editing Inversive geometry (section)

=== Properties ===
<gallery>
File:Inversion illustration2.svg|The inverse, with respect to the red circle, of a circle going through ''O'' (blue) is a line not going through ''O'' (green), and vice versa.
File:Inversion illustration3.svg|The inverse, with respect to the red circle, of a circle ''not'' going through ''O'' (blue) is a circle not going through ''O'' (green), and vice versa.
File:Inversion.gif|Inversion with respect to a circle does not map the center of the circle to the center of its image
</gallery>

The inversion of a set of points in the plane with respect to a circle is the set of inverses of these points.  The following properties make circle inversion useful.
*A circle that passes through the center ''O'' of the reference circle inverts to a line not passing through ''O'', but parallel to the tangent to the original circle at ''O'', and vice versa; whereas a line passing through ''O'' is inverted into itself (but not pointwise invariant).<ref name="Kay 1969 265">{{harvtxt|Kay|1969|p=265}}</ref>
*A circle not passing through ''O'' inverts to a circle not passing through ''O''. If the circle meets the reference circle, these invariant points of intersection are also on the inverse circle. A circle (or line) is unchanged by inversion [[if and only if]] it is [[Orthogonal circles|orthogonal]] to the reference circle at the points of intersection.<ref name="Kay 1969 265"/>

Additional properties include:
*If a circle ''q'' passes through two distinct points A and A' which are inverses with respect to a circle ''k'', then the circles ''k'' and ''q'' are orthogonal.
*If the circles ''k'' and ''q'' are orthogonal, then a straight line passing through the center O of ''k'' and intersecting ''q'', does so at inverse points with respect to ''k''.
*Given a triangle OAB in which O is the center of a circle ''k'', and points A' and B' inverses of A and B with respect to ''k'', then
:: <math> \angle OAB = \angle OB'A' \ \text{ and }\ \angle OBA = \angle OA'B'.</math>
*The points of intersection of two circles ''p'' and ''q'' orthogonal to a circle ''k'', are inverses with respect to ''k''.
*If M and M' are inverse points with respect to a circle ''k'' on two curves m and m', also inverses with respect to ''k'', then the tangents to m and m' at the points M and M' are either perpendicular to the straight line MM' or form with this line an isosceles triangle with base MM'.
*Inversion leaves the measure of angles unaltered, but reverses the orientation of oriented angles.<ref>{{harvtxt|Kay|1969|p=269}}</ref>