Let $g_1$ and $g_2$ be some lines, which intersect in point $A$. A circle $k_1$ is tangent to $g_1$ at point $A$ and intersects $g_2$ for a second time in $C$. A circle $k_2$ is tangent to $g_2$ at point $A$ and intersects $g_1$ for a second time in $D$. The circles $k_1$ and $k_2$ intersect for a second time in point $B$. Prove that, if $\frac{AC}{AD}=\sqrt{2}$, then $\frac{BC}{BD}=2$.