Say we have a line $L: ax+by+c=0 $ in the plane. Define the inversion of a point $(x,y)$ as $$\frac{(x,y)}{x^2+y^2}= (x',y')$$ with $$(x,y) \cdot (x',y')= 1.$$
What is the equation for the inversion of the line L?
I tried plugging in the equation for y but that didn't help...
We have $x=x'/(x'^2+y'^2)$ and $y=y'/(x'^2+y'^2)$. Then $ax+by+c=0$ becomes $$\frac{ax'+by'}{x'^2+y'^2}+c=0.$$ I'll leave simplification of this as an "exercise for the reader".