Editing Inversive geometry (section)

=== Application ===
For a circle not passing through the center of inversion, the center of the circle being inverted and the center of its image under inversion are [[collinear]] with the center of the reference circle. This fact can be used to prove that the [[Euler line]] of the [[intouch triangle]] of a triangle coincides with its OI line. The proof roughly goes as below:

Invert with respect to the [[incircle]] of triangle ''ABC''. The [[medial triangle]] of the intouch triangle is inverted into triangle ''ABC'', meaning the circumcenter of the medial triangle, that is, the nine-point center of the intouch triangle, the incenter and circumcenter of triangle ''ABC'' are [[collinear]].

Any two non-intersecting circles may be inverted into [[concentric]] circles. Then the [[inversive distance]] (usually denoted δ) is defined as the [[natural logarithm]] of the ratio of the radii of the two concentric circles.

In addition, any two non-intersecting circles may be inverted into [[congruence (geometry)|congruent]] circles, using circle of inversion centered at a point on the [[circle of antisimilitude]].

The [[Peaucellier–Lipkin linkage]] is a mechanical implementation of inversion in a circle. It provides an exact solution to the important problem of converting between linear and circular motion.

==== Pole and polar ====
{{main article|pole and polar}}
[[File:Pole and polar.svg|thumb|right|The polar line ''q'' to a point '''Q''' with respect to a circle of radius ''r'' centered on the point '''O'''.  The point '''P''' is the [[inversive geometry#Circle inversion|inversion point]] of '''Q'''; the polar is the line through '''P''' that is perpendicular to the line containing '''O''', '''P''' and '''Q'''.]]

If point ''R'' is the inverse of point ''P'' then the lines [[perpendicular]]  to the line ''PR'' through one of the points is the [[pole and polar|polar]] of the other point (the [[pole and polar|pole]]).

Poles and polars have several useful properties:

* If a point '''P''' lies on a line ''l'', then the pole '''L''' of the line ''l'' lies on the polar ''p'' of point '''P'''.
* If a point '''P''' moves along a line ''l'', its polar ''p'' rotates about the pole '''L''' of the line ''l''.
* If two tangent lines can be drawn from a pole to the circle, then its polar passes through both tangent points.
* If a point lies on the circle, its polar is the tangent through this point.
* If a point '''P''' lies on its own polar line, then '''P''' is on the circle.
* Each line has exactly one pole.