[The product of the lengths of the segments of chords and the lengths of the segments of secants] [Properties and characteristics of an isosceles triangle.] [Thales' theorem and the theorem of proportional segments]

On the extension of side $A D$ of rhombus $A B C D$ beyond point $D$, point $K$ is taken. Lines $A C$ and $B K$ intersect at point $Q$. It is known that $A K=14$ and that points $A, B$ and $Q$ lie on a circle with radius 6, the center of which lies on segment $A A$. Find $B K$.
Let $O$ be the center of the circle. Use the similarity of triangles $K Q O$ and $K B A$.

## Solution

Let $O$ be the center of the circle, $F$ be the second intersection point of the circle with the line $A$. Since triangle $A O Q$ is isosceles $(O A=O Q=6)$, then

$\angle A Q O=\angle O A Q=\angle B A Q$. Therefore, $O Q \| A B$. Hence, $K Q: K B=K O: K A=4: 7,4 / 7 K B^{2}=K Q \cdot K B=K F \cdot K A=$ 28, from which $B K=7$.

![](https://cdn.mathpix.com/cropped/2024_05_06_a7c46f4bbe36942e4ae1g-37.jpg?height=432&width=783&top_left_y=1&top_left_x=643)

## Answer

7.