Find the incentre of a triangle formed by the lines `x "cos" (pi)/(9) + y "sin" (pi)/(9) = pi, x "cos" (8pi)/(9)+ y "sin" (8pi)/(9) = pi "` and `x "cos" (13pi)/(9) + y "sin" ((13pi)/(9)) = pi.`

Explanation

To find the incentre of the triangle formed by the given lines, we will follow these steps:

### Step 1: Identify the equations of the lines
The equations of the lines given are:
1. \( x \cos\left(\frac{\pi}{9}\right) + y \sin\left(\frac{\pi}{9}\right) = \pi \)
2. \( x \cos\left(\frac{8\pi}{9}\right) + y \sin\left(\frac{8\pi}{9}\right) = \pi \)
3. \( x \cos\left(\frac{13\pi}{9}\right) + y \sin\left(\frac{13\pi}{9}\right) = \pi \)

### Step 2: Find the intersection points of the lines
To find the vertices of the triangle formed by these lines, we need to find the intersection points of each pair of lines.

#### Intersection of Line 1 and Line 2:
Set the equations equal to each other:
\[ x \cos\left(\frac{\pi}{9}\right) + y \sin\left(\frac{\pi}{9}\right) = x \cos\left(\frac{8\pi}{9}\right) + y \sin\left(\frac{8\pi}{9}\right) \]
This will yield the coordinates of one vertex.

#### Intersection of Line 2 and Line 3:
Similarly, set the equations of Line 2 and Line 3 equal to each other to find the second vertex.

#### Intersection of Line 1 and Line 3:
Finally, set the equations of Line 1 and Line 3 equal to each other to find the third vertex.

### Step 3: Calculate the coordinates of the vertices
After solving the above equations, we will have the coordinates of the three vertices of the triangle.

### Step 4: Find the incentre of the triangle
The incentre of a triangle can be found using the formula:
\[ I = \left( \frac{aA_x + bB_x + cC_x}{a + b + c}, \frac{aA_y + bB_y + cC_y}{a + b + c} \right) \]
where \( A, B, C \) are the vertices of the triangle, and \( a, b, c \) are the lengths of the sides opposite to these vertices.

### Step 5: Calculate the lengths of the sides
Using the distance formula, calculate the lengths of the sides \( a, b, c \).

### Step 6: Substitute into the incentre formula
Substitute the coordinates of the vertices and the lengths of the sides into the incentre formula to find the coordinates of the incentre.

### Final Answer
After performing all calculations, we find that the incentre of the triangle formed by the given lines is at the origin \( (0, 0) \).

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