Inversive geometry

Inversive geometry studies properties preserved by inversion in circles or spheres. Inversion exchanges the inside and outside of a circle while fixing its boundary.

Inversion (geometry): A transformation that sends each point to a reciprocal position along its ray from a fixed center. This transformation supplies the central operation whose invariant properties define the subject.
Euclidean plane: The flat two-dimensional space modeled by ordinary points, lines, distances, and angles. Planar inversion is defined through distances and rays in this familiar setting.
Problem of Apollonius: The classical problem of constructing circles tangent to three given circles or lines. Choosing an inversion centered at a tangency can reduce curved constraints to simpler line-and-circle configurations.
Reflection (mathematics): An isometry that flips points across a line, plane, or other specified mirror. Unlike reflection, inversion changes distances according to reciprocal radius while fixing a circle.
Circle: The set of points in a plane at a fixed distance from a center. The circle of inversion determines the reciprocal distances and remains pointwise fixed.
Circle inversion formula: The rule mapping a point at distance r from a center to the same ray at distance R²/r. It gives the exact reciprocal-distance relation behind every circle inversion.
Steiner chain: A cyclic chain of circles, each tangent to its neighbors and to two fixed circles. Inversion converts suitable coaxial boundary circles into parallel lines, clarifying the chain’s closure condition.
Similarity (geometry): A transformation that preserves shapes and angles while multiplying all distances by one constant factor. Inversion preserves angles but does not scale all distances by a single constant.
Cross-ratio: A ratio of directed distances among four points on a line or circle that is preserved by Möbius transformations. Its invariance captures a deeper structure shared by inversion and related transformations.
Power of a point: For a point and a circle, the product of its distances to the two intersections of any secant through it. The fixed product in inversion is the signed power of the point with respect to the circle.

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Casey's theoremRelated: Its transformations preserve circle tangency, a defining feature of Casey's configurations.