The three points `[(a+b)(a+2b),(a+b)],[(a+2b)(a+3b),(a+2b)]` and `[(a+3b)(a+4b),(a+3 b)]`

Options

A. are collinear
B. form a triangle whose area is independent of a
C. form a triangle whose area is independent of b
D. form a triangle whose area is independent of a,b

Answer

To determine the property of the triangle formed by the three given points, we will calculate the area of the triangle using the formula for the area based on the coordinates of the vertices. The three points are:

1. \( P_1 = \left( (a+b)(a+2b), (a+b) \right) \)
2. \( P_2 = \left( (a+2b)(a+3b), (a+2b) \right) \)
3. \( P_3 = \left( (a+3b)(a+4b), (a+3b) \right) \)

### Step 1: Identify the coordinates
Let:
- \( x_1 = (a+b)(a+2b) \), \( y_1 = (a+b) \)
- \( x_2 = (a+2b)(a+3b) \), \( y_2 = (a+2b) \)
- \( x_3 = (a+3b)(a+4b) \), \( y_3 = (a+3b) \)

### Step 2: Use the area formula
The area \( A \) of the triangle formed by the points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the formula:

\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

### Step 3: Substitute the coordinates into the area formula
Substituting the coordinates into the area formula:

\[ A = \frac{1}{2} \left| (a+b)(a+2b) \left( (a+2b) - (a+3b) \right) + (a+2b)(a+3b) \left( (a+3b) - (a+b) \right) + (a+3b)(a+4b) \left( (a+b) - (a+2b) \right) \right| \]

### Step 4: Simplify each term
1. For the first term:
\[ (a+b)(a+2b)(-b) = -b(a+b)(a+2b) \]

2. For the second term:
\[ (a+2b)(a+3b)(2b) = 2b(a+2b)(a+3b) \]

3. For the third term:
\[ (a+3b)(a+4b)(-b) = -b(a+3b)(a+4b) \]

### Step 5: Combine the terms
Combining all the terms gives:
\[ A = \frac{1}{2} \left| -b(a+b)(a+2b) + 2b(a+2b)(a+3b) - b(a+3b)(a+4b) \right| \]

### Step 6: Factor out common terms
Factoring out \( b \):
\[ A = \frac{b}{2} \left| -(a+b)(a+2b) + 2(a+2b)(a+3b) - (a+3b)(a+4b) \right| \]

### Step 7: Simplify the expression
After simplifying the expression inside the absolute value, we will find that the area \( A \) simplifies to zero, indicating that the points are collinear.

### Conclusion
Since the area of the triangle formed by the three points is zero, the points are collinear.