= Solution
For a <holomorphic function>, the <Cauchy-Riemann equations> are $u_x=v_y$ and $u_y=-v_x$. Consequently
$$
\boxed{\nabla u\cdot\nabla v=u_xv_x+u_yv_y=0,\qquad |\nabla u|^2=u_x^2+u_y^2=v_x^2+v_y^2=|\nabla v|^2.}
$$
A <gradient> is normal to its <level set>, so these identities give orthogonal level curves wherever the gradients are nonzero. At a critical point where $f'=0$, both gradients vanish and a regular tangent to a <level set> need not exist; the geometric orthogonality statement is restricted to regular intersections.
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