Ex. 16 of Ch. 9 from the book "Differential Geometry", by A. V. Pogorelov, goes as follows:
"The surfaces $F_1$ and $F_2$ are called the surfaces of centers of the surface $F$ if they are formed by the endpoints of segments of lenghts $1/\kappa_1$ and $1/\kappa_2$ ($\kappa_1$ and $\kappa_2$ are the principal curvatures of $F$), marked off on the normals to the surface $F$. A point correspondence is estalished in a natural way between the surfaces $F_1$, $F_2$ and $F$. Namely points on the surfaces lying on the same normal to F are corresponding points. Prove that geodesic curves on the surfaces of centers correspond to lines of curvatures on the surface $F$."
Some definitions and known results:
A line of curvature is a curve on a surface such that its direction at every point is a principal direction.
A geodesic curve $\gamma$ on $X$ is a curve that satisfies $\gamma''//n$, where $n$ is the normal vector field associated to $X$.
Given a curve $\gamma$ on a surface $X$, its geodesic curvature is $\kappa_g(\gamma) := \gamma'' \cdot (n \times \gamma')$.
If $\gamma$ is a geodesic curve, then its geodesic curvature is equal to zero.
Locally, geodesics minimize distances on a surface.
My ideia was: Given a parametrization $r(u,v)$ of a neighborhood on $F$, we can induce parametrizations $r_1, r_2$ on $F_1$ and $F_2$, namely \begin{align*} r_1(u,v) = r(u,v) + \frac{1}{\kappa_1(u,v)}n(u,v), \\ r_2(u,v) = r(u,v) + \frac{1}{\kappa_2(u,v)}n(u,v). \end{align*} My intuition says I should find some relation between $n_1, n_2$ and $n$, and use the geodesic curvature result to get something, but I got nowhere. Maybe there's some special parametrization that would make things simpler, but I'm not sure. Does anybody have a clue on how to solve this?
(Of course, I'm always considering neighborhoods on $F$ such that both the principal curvatures are non-zero, so that $F_1$ and $F_2$ are well-defined)