If (x, y) is the incentre of the triangle formed by the points (3, 4), (4, 3) and (1, 2), then the value of `x^(2)` is

Answer

To find the value of \( x^2 \) where \( (x, y) \) is the incenter of the triangle formed by the points \( (3, 4) \), \( (4, 3) \), and \( (1, 2) \), we can follow these steps:

### Step 1: Identify the vertices of the triangle
Let the vertices of the triangle be:
- \( A(3, 4) \)
- \( B(4, 3) \)
- \( C(1, 2) \)

### Step 2: Calculate the lengths of the sides of the triangle
We will use the distance formula to calculate the lengths of the sides opposite to each vertex.

1. **Length of side \( a \) (opposite vertex A)**:
\[ a = BC = \sqrt{(4 - 1)^2 + (3 - 2)^2} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \]

2. **Length of side \( b \) (opposite vertex B)**:
\[ b = AC = \sqrt{(3 - 1)^2 + (4 - 2)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

3. **Length of side \( c \) (opposite vertex C)**:
\[ c = AB = \sqrt{(4 - 3)^2 + (3 - 4)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Use the incenter formula to find coordinates \( (x, y) \)
The coordinates of the incenter \( (x, y) \) can be calculated using the formula:
\[ x = \frac{a x_1 + b x_2 + c x_3}{a + b + c} \]
\[ y = \frac{a y_1 + b y_2 + c y_3}{a + b + c} \]

Substituting the values:
- \( x_1 = 3, y_1 = 4 \)
- \( x_2 = 4, y_2 = 3 \)
- \( x_3 = 1, y_3 = 2 \)

Calculating \( x \):
\[ x = \frac{\sqrt{10} \cdot 3 + 2\sqrt{2} \cdot 4 + \sqrt{2} \cdot 1}{\sqrt{10} + 2\sqrt{2} + \sqrt{2}} \]
\[ = \frac{3\sqrt{10} + 8\sqrt{2} + \sqrt{2}}{\sqrt{10} + 3\sqrt{2}} \]
\[ = \frac{3\sqrt{10} + 9\sqrt{2}}{\sqrt{10} + 3\sqrt{2}} \]

### Step 4: Calculate \( x^2 \)
To find \( x^2 \):
\[ x^2 = \left( \frac{3\sqrt{10} + 9\sqrt{2}}{\sqrt{10} + 3\sqrt{2}} \right)^2 \]

This expression can be simplified further, but we can directly find \( x^2 \) by substituting the values we have already calculated.

### Final Calculation
After evaluating the above expression, we find:
\[ x = 3 \implies x^2 = 9 \]

### Final Answer
Thus, the value of \( x^2 \) is \( \boxed{9} \).

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