Prove Analytically: Inversion of a Circle is Also a Circle

[zhandele]

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.

 

[zhandele]

I'm posting again to upload pdf files I generated from Geometer's Sketch Pad. They may help somebody follow what I write in the other two pdfs. Thanks!