Find the incentre of the triangle with vertices `(1, sqrt3), (0, 0)` and `(2, 0)`

Explanation

To find the incenter of the triangle with vertices \( A(1, \sqrt{3}) \), \( B(0, 0) \), and \( C(2, 0) \), we will follow these steps:

### Step 1: Identify the vertices of the triangle
The vertices of the triangle are:
- \( A(1, \sqrt{3}) \)
- \( B(0, 0) \)
- \( C(2, 0) \)

### Step 2: Calculate the lengths of the sides of the triangle
Using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), we calculate the lengths of the sides \( AB \), \( BC \), and \( CA \).

1. **Length of \( AB \)**:
\[ AB = \sqrt{(1 - 0)^2 + (\sqrt{3} - 0)^2} = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

2. **Length of \( BC \)**:
\[ BC = \sqrt{(2 - 0)^2 + (0 - 0)^2} = \sqrt{2^2 + 0^2} = \sqrt{4} = 2 \]

3. **Length of \( CA \)**:
\[ CA = \sqrt{(1 - 2)^2 + (\sqrt{3} - 0)^2} = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

### Step 3: Determine if the triangle is equilateral
Since \( AB = BC = CA = 2 \), the triangle is equilateral.

### Step 4: Find the centroid of the triangle
For an equilateral triangle, the incenter coincides with the centroid. The formula for the centroid \( G \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

Substituting the coordinates of the vertices:
\[ G = \left( \frac{1 + 0 + 2}{3}, \frac{\sqrt{3} + 0 + 0}{3} \right) = \left( \frac{3}{3}, \frac{\sqrt{3}}{3} \right) = (1, \frac{\sqrt{3}}{3}) \]

### Step 5: Conclusion
The incenter of the triangle is:
\[ \boxed{\left( 1, \frac{\sqrt{3}}{3} \right)} \]