Prove Analytically: Inversion of a Circle is Also a Circle

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SUMMARY

The discussion centers on proving analytically that the inversion of a circle in the unit circle remains a circle. The unit circle is defined by the equation x² + y² = 1, while the general circle D is represented by (x-a)² + (y-b)² = c², with the condition that the center does not coincide with the origin (a² + b² ≠ c²). Participants share their attempts at deriving the equation of the image circle D' and express challenges in the algebraic proof, despite successfully demonstrating the concept synthetically.

PREREQUISITES
  • Understanding of circle equations in the Euclidean plane
  • Familiarity with the concept of inversion in geometry
  • Knowledge of synthetic versus analytic proofs
  • Experience with Geometer's Sketch Pad (GSP) for visualizing geometric concepts
NEXT STEPS
  • Study the properties of circle inversion in Euclidean geometry
  • Learn how to derive the equation of the inverted circle analytically
  • Explore the use of Geometer's Sketch Pad (GSP) for geometric proofs
  • Investigate additional examples of geometric transformations and their properties
USEFUL FOR

This discussion is beneficial for geometry students, educators, and mathematicians interested in geometric transformations, particularly those focusing on circle inversion and its properties.

zhandele
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Homework Statement



Given the unit circle (in the Euclidean plane) centered at the origin x^2+y^2=1, and a general circle D with equation (x-a)^2+(y-b)^2=c^2 that does not pass through the origin (ie the center of inversion, ie a^2+b^2≠c^2, prove analytically that the inversion of D in the unit cirlce is still a circle.

Homework Equations



See the attached pdf files

The Attempt at a Solution



I can prove this synthetically. I even worked out the equation of the image circle D', but I can't derive it algebraically. I feel I must be missing something very obvious.

I uploaded two pdf files. I was going to upload a GSP file, but I guess this forum can't do that? I'll have to generate pdfs from it or something. Do most of you guys have GSP? It's mind-bogglingly useful.
 

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