The vertices of a triangle are (a, b - c), (b, c - a) and (c, a - b). Prove that its centroid lies on x-axis.

Answer

To prove that the centroid of the triangle with vertices at (a, b - c), (b, c - a), and (c, a - b) lies on the x-axis, we will follow these steps:

### Step 1: Identify the vertices of the triangle
The vertices of the triangle are given as:
- Vertex A: \( (a, b - c) \)
- Vertex B: \( (b, c - a) \)
- Vertex C: \( (c, a - b) \)

### Step 2: Use the formula for the centroid
The coordinates of the centroid \( G \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

### Step 3: Substitute the coordinates of the vertices into the centroid formula
Substituting the coordinates of the vertices into the centroid formula:
- \( x_1 = a \), \( y_1 = b - c \)
- \( x_2 = b \), \( y_2 = c - a \)
- \( x_3 = c \), \( y_3 = a - b \)

The x-coordinate of the centroid \( G \) is:
\[ x_G = \frac{a + b + c}{3} \]

The y-coordinate of the centroid \( G \) is:
\[ y_G = \frac{(b - c) + (c - a) + (a - b)}{3} \]

### Step 4: Simplify the y-coordinate
Now, let's simplify the y-coordinate:
\[ y_G = \frac{(b - c) + (c - a) + (a - b)}{3} \]
Combining the terms:
\[ y_G = \frac{b - c + c - a + a - b}{3} \]
Notice that \( b - b = 0 \), \( c - c = 0 \), and \( a - a = 0 \):
\[ y_G = \frac{0}{3} = 0 \]

### Step 5: Conclusion
Since the y-coordinate of the centroid \( G \) is 0, we conclude that the centroid lies on the x-axis.

### Final Answer
Thus, the centroid of the triangle with vertices at \( (a, b - c) \), \( (b, c - a) \), and \( (c, a - b) \) lies on the x-axis.

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