Area of triangle having angular points
`A[am_1m_2, (m_1+m_2) ], B[am_2m_3, a(m_2+m_3)` and `C[am_3m_1. a(m_3+ m_1) ]` is

Options

A. ` (m_1-m_2) (m_2-m_3) (m_3-m_1) `
B. ` a^(2) (m_1-m_2) (m_2-m_3) (m_3-m_1)`
C. ` (1)/(2) a^(2) (m_1-m_2) (m_2-m_3) (m_3-m_1) `
D. none of these

Answer

To find the area of the triangle with vertices A, B, and C given by the coordinates:

- A \((am_1m_2, (m_1+m_2))\)
- B \((am_2m_3, a(m_2+m_3))\)
- C \((am_3m_1, a(m_3+m_1))\)

we will use the formula for the area of a triangle given by its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

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

### Step-by-Step Solution:

1. **Identify the Coordinates:**
- Let \(A = (x_1, y_1) = (am_1m_2, m_1 + m_2)\)
- Let \(B = (x_2, y_2) = (am_2m_3, a(m_2 + m_3))\)
- Let \(C = (x_3, y_3) = (am_3m_1, a(m_3 + m_1))\)

2. **Substitute the Coordinates into the Area Formula:**
\[ \text{Area} = \frac{1}{2} \left| am_1m_2(a(m_2 + m_3) - a(m_3 + m_1)) + am_2m_3(a(m_3 + m_1) - (m_1 + m_2)) + am_3m_1((m_1 + m_2) - a(m_2 + m_3)) \right| \]

3. **Simplify Each Term:**
- For the first term:
\[ am_1m_2(a(m_2 + m_3) - a(m_3 + m_1)) = am_1m_2(a(m_2 - m_1)) \]
- For the second term:
\[ am_2m_3(a(m_3 + m_1) - (m_1 + m_2)) = am_2m_3(a(m_3 - m_2)) \]
- For the third term:
\[ am_3m_1((m_1 + m_2) - a(m_2 + m_3)) = am_3m_1(m_1 + m_2 - a(m_2 + m_3)) \]

4. **Combine the Terms:**
\[ \text{Area} = \frac{1}{2} \left| a \left( m_1m_2(m_2 - m_1) + m_2m_3(m_3 - m_2) + m_3m_1(m_1 + m_2 - a(m_2 + m_3)) \right) \right| \]

5. **Factor Out Common Terms:**
- Factor out \(a\) from the expression to simplify:
\[ \text{Area} = \frac{a}{2} \left| m_1m_2(m_2 - m_1) + m_2m_3(m_3 - m_2) + m_3m_1(m_1 + m_2 - a(m_2 + m_3)) \right| \]

6. **Final Expression for Area:**
- The final expression for the area of the triangle is:
\[ \text{Area} = \frac{a}{2} \left| m_1m_2(m_2 - m_1) + m_2m_3(m_3 - m_2) + m_3m_1(m_1 + m_2 - a(m_2 + m_3)) \right| \]