$\left.\begin{array}{l}\text { [ Law of Cosines } \\ \text { [ Angles subtending equal arcs and equal chords }]\end{array}\right]$

An equilateral triangle $ABC$ with a side length of 3 is inscribed in a circle. Point $D$ lies on the circle, and the chord $AD$ is equal to $\sqrt{3}$. Find the chords $BD$ and $CD$.
Using the Law of Cosines for triangle $A D B$, write an equation in terms of $B D$.

## Solution

Assume that point $D$ lies on the arc $A B$ not containing point $C$. In triangle $A D B$, it is known that

$$
A D=\sqrt{3}, A B=3, \angle A D B=120^{\circ}
$$

Let $D B=x$ and write the Law of Cosines for this triangle:

$$
A B^{2}=A D^{2}+B D^{2}-2 A D \cdot B D \cos \angle A D B
$$

or

$$
9=3+x^{2}+x \sqrt{3} \text {, or } x^{2}+x \sqrt{3}-6=0 .
$$

From this, we find that $x=\sqrt{3}$.

In triangle $A D C$, it is known that

$$
A D=\sqrt{3}, A C=3, \angle A D C=\angle A B C=60^{\circ} .
$$

Similarly, we find that $C D=2 \sqrt{3}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_16ab69fab1743849dba9g-15.jpg?height=463&width=481&top_left_y=2038&top_left_x=797)

Answer

$\sqrt{3}, 2 \sqrt{3}$ or $2 \sqrt{3}, \sqrt{3}$.