# The Use of Inversion in Classical Geometry

- Author:
- Christine Liu

The Use of Inversion in Classical Geometry

## Table of Contents

### Circle Inversion

- Definition of circle inversion
- ΔOAB ~ ΔOB’A’
- Different cases for circle inversion of circles and lines
- Objects that are tangent to each other, are inverted to objects that are also tangent to each other.
- If a circle d is inverted in another circle c, then d is inverted to itself if and only if c and d are perpendicular.

### Ptolemy's Theorem

### Nine-Point Circle

- In any triangle, the mid-points of the sides, the feet of altitudes, and the mid-points of the segments that join the orthocentre H with each vertex all lie on a common circle.
- The centre of the nine-point circle lies on the Euler line and is the mid-point of the circumcentre O and the orthocentre H.
- Feuerbach's Theorem
- Feuerbach's Theorem proof using circle inversion
- Nine-Point Conic
- Generalization of Euler Line
- Lester's Theorem

### Apollonius' Problem

### Peaucellier linkage