I have a question. How can i show a vector bisect a angle $\omega$ between coordinate curves in a regular surface $S$ ?
The problem: Let $X(u,v)$ a local parametrization of a surface $S$. Show that a vector $\alpha X_{u}+ \beta X_{v}$ bisect a angle between coordinate curves if and only if $\sqrt{G} (\bigl((\alpha E + \beta F)\bigr)$ $=$ $\sqrt{E} (\bigl((\alpha F + \beta G)\bigr)$.
I know that vector $\alpha X_{u} + \beta X_{v}$ belong to $T_{p} S$ and the angle $\omega$ between coordinate curves is:
$cos \omega$ $=$ $\frac{<X_{u},X_{v}>}{\Vert{X_{u}}\Vert \Vert{X_{v}}\Vert}$
but i don't understand how can i show bisection.