50. Given a triangle $ABC$. The tangent to the circumcircle of this triangle at point $B$ intersects the line $AC$ at point $M$. Find the ratio $|AM|:|MC|$, if $|AB|:|BC|=k$.
50. We have

$$
\begin{aligned}
\frac{|M A|}{|M C|} & =\frac{S_{A B M}}{S_{M B C}}=\frac{\frac{1}{2}|M B| \cdot |R A| \sin \widehat{M B A}}{\frac{1}{2}|M B| \cdot|B C| \sin \widehat{M B C}}= \\
& =\frac{|B A|^{2}}{|B C|^{2}} \frac{|B C|}{\sin \widehat{M B C}} \frac{\sin \widehat{M B A}}{|B A|}
\end{aligned}
$$

But $\sin \widehat{M B C}=\sin \widehat{B A C}, \sin \widehat{M B A}=\sin \widehat{B C A}$, and by the Law of Sines $\frac{|B C|}{\sin \widehat{B A C}}=\frac{|B A|}{\sin \widehat{B C A}}$. Therefore, $\frac{|M A|}{|M C|}=\frac{|B A|^{2}}{|B C|^{2}}=k^{2}$