Inversion in a Circle
A map sending P to P′ with OP·OP′ = r².
Key idea
A map sending P to P′ with OP·OP′ = r².
In depth
Inversion about a circle of radius r sends a point P to P′ on the same ray with OP · OP′ = r². Circles and lines map to circles and lines, and it turns hard configurations into easy ones.
Worked example
With r = 2, a point at distance 4 inverts to distance 1.
Real-world uses
- Advanced geometry used in complex analysis and in solving difficult circle problems
Common mistake
Forgetting that the centre has no image.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Inversion in a Circle?
A map sending P to P′ with OP·OP′ = r². It is a topic at the expert level in Geometry.
Where is Inversion in a Circle used in real life?
Real-world examples: Advanced geometry used in complex analysis and in solving difficult circle problems.
What should I learn before Inversion in a Circle?
Make sure you are comfortable with Mass Point Geometry, the previous topic in Geometry.
What is a common mistake with Inversion in a Circle?
Forgetting that the centre has no image.
How can I practise Inversion in a Circle?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Inversion in a Circle?
Move on to Area and Volume Scaling next.
Related expert topics in Geometry
- Power of a Point – For a point and a circle, products of segment lengths along secants are equal.
- Cyclic Quadrilaterals – Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
- Cavalieri’s Principle – Solids with equal cross-sectional areas at every height have equal volume.
- Area by Coordinates (Shoelace) – Area of a polygon from vertices.
- Parabola (Geometric Definition) – Points equidistant from a focus and a directrix.
- Ellipse – Sum of distances to two foci is constant.