Tangencies: Inversion

Inversion is a very useful symmetry operation on circles, generalizing mirror reflection through a line. An inversion is performed with respect to a particular center point and choice of scale; then each point in the plane is transformed to another point, forming the same angle to the center but with distance inversely proportional to its original distance.

The center and scale can be specified as a single circle (shown in red in the animation below). To invert a point outside this circle, such as the blue one in the bottom right of the animation, draw a circle (yellow) having as its diameter the segment from the given point to the inversion circle's center; this circle crosses the inversion circle in two points; connect those points by a line and also draw the perpendicular line from the given point to the inversion center (also yellow). These two lines cross at the inverse of the point. To invert a point inside the circle, simply reverse these steps. The animation below depicts several geometric figures (blue) and their images (green) under inversion through the red circle.

If you were running Java, you'd see a nice animation instead of this gif.

The key properties of inversion are that it transforms circles to circles, and preserves the angles of crossings between circles; in particular inversion preserves the tangencies of a collection of circles, such as the three tangent circles shown in the animation. Note that the blue line through two points is inverted into a circle, through the center of the inversion. When interpreting geometric figures using inversion, lines should be viewed as infinite-radius circles that go through a special "point at infinity" which is the image under inversion of the inversion circle's center.

Many geometric constructions and proofs can be simplified by using inversion to transform the figure into a more symmetric situation.

Animation created by Cinderella.
From the Geometry Junkyard, computational and recreational geometry.
David Eppstein, Theory Group, ICS, UC Irvine.

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