Trying to wrap my head around inversions. I understand it takes things from inside to outside, such that
$\text{distance from some point inside circle} + \text{ distance to new point}=r^2$
Where $r$ is radius of circle of inversion.
How would this look with a figure not just one line? For example a triangle inscribed in a circle? I know angles are supposed to be preserved.
Let $O$ be the centre of the circle, $P$ our point "inside" and $Q$ the point it s mapped to. Then $(OP)(OQ)=r^2$. (Your expression in terms of sums is not quite right.)
As to your question about the triangle, in general lines are taken to circles, except that lines through the origin are taken to themselves. So if the extensions of the lines that make up your triangle don't go through the origin, then the sides of the triangle will be sent to arcs of circles. These arcs will make up a vaguely triangular shape, but with circular arcs rather than straight lines. However, these arcs meet at angles that match the original angles of the triangle.