51. On a straight line, points $A, B, C$, and $D$ are sequentially arranged, such that $|A C|=\alpha|A B|, \quad|A D|=\beta|A B|$. A circle is drawn through $A$ and $B$, and $C M$ and $D N$ are two tangents to this circle ( $M$ and $N$ are points on the circle, lying on opposite sides of the line $A B$ ). In what ratio does the line $M N$ divide the segment $A B$?
51. From problem 50, it follows that \(\frac{|A M\rangle^{2}}{|M B|^{2}}=\frac{|A C|}{|B C|}, \frac{|A N|^{2}}{|N B|^{2}}=\frac{|A D|}{|B D|^{2}}\). If \(K\) is the intersection point of \(M N\) and \(A B\), then

\[
\begin{aligned}
\frac{|A K|}{|K B|} & =\frac{S_{A M N}}{S_{B M N}}=\frac{|A M| \cdot|A N| \sin \widehat{M A N}}{|M B| \cdot|N B| \sin \widehat{M B N}}= \\
& =\sqrt{\frac{|A C|}{|B C|} \cdot \frac{|A D|}{|B D|}}=\sqrt{\frac{\alpha \beta}{(\alpha-1)(\beta-1)}}
\end{aligned}
\]