9.1. (12 points) Two circles touch each other externally at point $K$. On their common internal tangent, point $P$ is marked such that $K P=14$. Through point $P$, two secants are drawn to the circles, with one of them intercepting a chord $A B=45$ on the first circle, and the other intercepting a chord $C D=21$ on the second circle, with point $A$ lying between points $B$ and $P$, and point $C$ lying between points $D$ and $P$. Find the ratio $B C: A D$.
Answer: 1.75.

Solution. By the tangent-secant theorem, we find $A P=4, C P=7$. Moreover, $P A \cdot P B=$ $P K^{2}=P C \cdot P D$, so triangles $B P C$ and $D P A$ are similar by two proportional sides and the angle between them. Therefore, $B C: A D=C P: A P=\frac{7}{4}$.