[Product of chord segment lengths and secant segment lengths] Diameter, main properties

Side $A B$ of a regular hexagon $A B C D E F$ is equal to $\sqrt{3}$ and is a chord of a certain circle, with the other sides of the hexagon lying outside this circle. The length of the tangent $C M$, drawn to the same circle from vertex $C$, is 3. Find the diameter of the circle.
Prove that the line $B C$ passes through the center of the given circle.

## Solution

Let $M$ be the point of tangency, $K$ be the second intersection point of the line $B C$ with the given circle (see figure). Then $C K \cdot C B = C M^2$, from which $B K = 2 \sqrt{3}$. In triangle $A B K$, $B K = 2 A B$, $\angle A B K = 180^{\circ} - \angle A B C = 180^{\circ} - 120^{\circ} = 60^{\circ}$. Therefore, triangle $A B K$ is a right triangle with hypotenuse $B K$. Thus, $B K = 2 \sqrt{3}$ is the desired diameter.

![](https://cdn.mathpix.com/cropped/2024_05_06_36c4956010bd91c4b4eeg-08.jpg?height=329&width=466&top_left_y=2437&top_left_x=796)

Problem