Geometry • Topic 28

Inversion

Inversion is a "reflection" across a circle. It turns "inside out," mapping the interior of a circle to the exterior. It is powerful because it preserves angles (conformal) and maps circles to lines/circles.

Definition

Definition (Inversion)

Given a circle

ω

with center

O

and radius

R

, the inverse of a point

P

(

P \neq = O

) is the point

P^{'}

on the ray

OP

such that:

OP \cdot O P^{'} = R^{2}

Key Mapping Properties

Lines through $O$ : Map to themselves (as lines).
Lines not through $O$ : Map to circles passing through $O$ .
Circles through $O$ : Map to lines not passing through $O$ .
Circles not through $O$ : Map to circles not passing through $O$ .

Distances

The distance formula under inversion:

A^{'} B^{'} = A B \cdot \frac{R ^{2}}{O A \cdot OB}

Applications

Ptolemy's Theorem: Inverting a quadrilateral leads to a simple proof.
Apollonius Problem: Constructing a circle tangent to 3 given circles becomes easier by inverting to turn circles into lines.

Practice Problems

Exercise (Problem 1)

Invert a square centered at

O

with respect to its circumcircle. What shape is formed?

Exercise (Problem 2)

Use inversion to prove that for four distinct points

A, B, C, D

, the cross-ratio is invariant under inversion.

Exercise (Problem 3)

Given two non-intersecting circles, show there exists an inversion that maps them to concentric circles.