[ The product of the lengths of the chord segments and the lengths of the secant segments ]

On a line, points $A, B, C$, and $D$ are arranged in the given order. It is known that $B C = 3, A B = 2 \cdot C D$. A circle is drawn through points $A$ and $C$, and another circle is drawn through points $B$ and $D$. Their common chord intersects segment $B C$ at point $K$. Find $B K$.

#
Apply the theorem of intersecting chords twice.

## Solution

Let $C D=a, A B=2 a$. Denote $B K=x$. Then

$$
K C=3-x, K D=D C+K C=a+3-x, A K=A B+B K=2 a+x
$$

Let $M N$ be the common chord of the specified circles. Then

$$
A K \cdot K C=M K \cdot N K=B K \cdot K D
$$

from which

From this equation, we find that $x=2$.

## Answer

2.