Picture below is from 5.2 Theorem of chapter 8 of do Carmo's Riemannian Geometry. I can't understand the red line. Why $f$ preserve angles means that $f$ map radii into radii ?
$U\subset \mathbb R^n$ is an open set. $f: U \rightarrow \mathbb R^n$ is a conformal transformation, but have not concrete expression. From equation (11), I have the diagrammatic drawing.
Besides, in my view, $f$ just be a conformal satisfy equation (11). But I'm not sure whether I'm missing something, so I add the all proof in pictures bottom.
