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Orthogonal families of curves

Let E, F, G be the coefficients of the first fundamental form of a regular surface S in the parametrization \mathbf{x}:U \subset \mathbb{R}^2 \rightarrow S. Let \varphi (u,v) = c and \psi(u,v)=d be two families of regular curves on \mathbf{x}(U) \subset S. Prove that these two families are orthogonal if and only if

E\varphi_v \psi_v-F(\varphi_u\psi_v+\varphi_v\psi_u) +G\varphi_u\psi_u=0

All I can see is splitting up the terms were we would get (\mathbf{x}_u\varphi_v-\mathbf{x}_v\varphi_u)(\mathbf{x}_u\psi_v-\mathbf{x}_v\psi_u)=0 but that does not go anywhere.

(attached is the tex file and it's pdf build since the latex editor is not rendering)Attachment 21477Attachment 21476