Question 61
Let $C_{1}$ and $C_{2}$ be two circles that are externally tangent at the point $P$. From a point $A$ of $C_{2}$ we draw two lines tangent to $C_{1}$ at the points $M$ and $M^{\prime}$. Let $N$ and $N^{\prime}$ be the respective points where these lines meet $C_{2}$, both different from $A$.
Prove that $\left|P N^{\prime}\right| \cdot|M N|=|P N| \cdot\left|M^{\prime} N^{\prime}\right|$.
Prove that $\left|P N^{\prime}\right| \cdot|M N|=|P N| \cdot\left|M^{\prime} N^{\prime}\right|$.