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Geometric Inversion

presented by Kyle Schultz

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This instructional unit was inspired by an instructional unit of the same title created by Pam Turner for EMAT 6690 in the spring of 1995. My intentions are to preserve Pam's work, to update it to reflect a decade's worth of technological advances, to improve it where possible, and to make it available to a wider audience. This unit uses two software packages: Geometer's Sketchpad and Graphing Calculator.

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A Note to the Teacher: The instructional unit that can be accessed through the links below contain suggestions for questions and topics intended for student exploration. These items will be underlined. Solutions and proofs for these questions, as well as notes for the teacher, are formatted in italics.

 

Introduction to Geometric Inversion

Conceptual Foundation

Analytic Geometry

 

Constructing Inverses

Introduction

Method 1: Proportions

Method 1a: Proportions

Method 2: Similar Triangles

Method 3: P Outside Gamma

Method 4: P Inside Gamma

Method 4a: P Inside Gamma

 

Inverting Geometric Figures

Introduction

Inverting Circles

Inverting Lines

Inverting More Circles

Inverting Other Conic Sections

 

The Analytic Geometry of Inverting Curves

Introduction

Curve #1: an ellipse centered at the origin

Curve #2: an ellipse not centered at the origin

Curve #3: a parabola with vertex at the origin

Curve #4: a parabola with vertex not at the origin

Curve #5: a hyperbola centered at the origin and tangent to gamma

Curve #6: a hyperbola centered at the origin

Curve #7: a hyperbola not centered at the origin

 

Stereographics

Introduction

Cross-Section of Construction

 

Applications

Introduction

1. Influence of Inversion on Segment Length

2a. Ptolemy's Theorem

2b. An Extension

2c. Fermat's Problem

2d. Cubic from an Inscribed Quadrilateral

3. Concentric Circles

4. Extended Diameter

5. Three Chords in a Circle

 

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