We are given an equilateral triangle $ABC$ with a circle of radius 1 that is tangent to the lines $AB$ and $AC$ at points $B$ and $C$, respectively. We need to find the side length of the triangle.

Step-by-step solution:

1. Understanding the configuration: Since the triangle is equilateral, all sides are equal, and all angles are 60 degrees. The circle is tangent to the sides $AB$ and $AC$, meaning the circle touches the sides at exactly one point (at $B$ and $C$, respectively). This is a classic example of an incircle of a triangle, but the circle is tangent to two of the sides, not all three.

2. Using the radius of the incircle: The formula for the radius $r$ of the incircle of an equilateral triangle with side length $s$ is:

   $r = \frac{s \sqrt{3}}{6}$

   The radius is given as 1. Therefore, we set the formula equal to 1:

   $\frac{s \sqrt{3}}{6} = 1$

3. Solving for the side length: To find $s$, we solve the equation:

   $s \sqrt{3} = 6$

   $s = \frac{6}{\sqrt{3}} = 2 \sqrt{3}$

Thus, the side length of the equilateral triangle $ABC$ is $2 \sqrt{3}$.

Would you like a detailed diagram to help visualize this? Let me know if you have any questions or need further clarification.

Related Questions:

1. What is the area of the equilateral triangle if the side length is $2 \sqrt{3}$?
2. How do you calculate the perimeter of an equilateral triangle given its side length?
3. Can the same method be used to calculate the incircle radius for other types of triangles (non-equilateral)?
4. What would happen to the radius of the incircle if the side length of the triangle is doubled?
5. How does the radius of an excircle differ from that of an incircle in an equilateral triangle?

Tip:

Always remember that the relationship between the side length and the radius of the incircle in an equilateral triangle is proportional to the square root of 3.