The sides of triangle ABC satisfy the relations `a + b - c= 2 and 2ab -c^(2) =4`, then the square of the area of triangle is ______

Explanation

To find the square of the area of triangle ABC given the conditions \( a + b - c = 2 \) and \( 2ab - c^2 = 4 \), we can follow these steps:

### Step 1: Use the first equation
From the first equation, we have:
\[ a + b - c = 2 \implies a + b = c + 2 \]

### Step 2: Square both sides
Now, we will square both sides of the equation \( a + b = c + 2 \):
\[ (a + b)^2 = (c + 2)^2 \]
Expanding both sides, we get:
\[ a^2 + 2ab + b^2 = c^2 + 4c + 4 \]

### Step 3: Rearrange the equation
Now, we can rearrange this equation:
\[ a^2 + b^2 + 2ab - c^2 - 4c - 4 = 0 \]

### Step 4: Substitute the second equation
We know from the second equation that:
\[ 2ab - c^2 = 4 \]
Substituting \( 2ab = c^2 + 4 \) into the rearranged equation gives:
\[ a^2 + b^2 + (c^2 + 4) - c^2 - 4c - 4 = 0 \]
This simplifies to:
\[ a^2 + b^2 - 4c = 0 \implies a^2 + b^2 = 4c \]

### Step 5: Use the identity for \( (a - b)^2 \)
We can also express \( a^2 + b^2 \) using the identity:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting \( a + b = c + 2 \) gives:
\[ a^2 + b^2 = (c + 2)^2 - 2ab \]
Substituting \( 2ab = c^2 + 4 \) into this equation leads to:
\[ a^2 + b^2 = (c + 2)^2 - (c^2 + 4) \]

### Step 6: Simplify the expression
Expanding \( (c + 2)^2 \):
\[ (c + 2)^2 = c^2 + 4c + 4 \]
Thus,
\[ a^2 + b^2 = c^2 + 4c + 4 - c^2 - 4 = 4c \]
This confirms our earlier result.

### Step 7: Establish relationships between sides
From \( a^2 + b^2 = 4c \) and \( 2ab - c^2 = 4 \), we can deduce that:
- If \( a = b = c \), then \( a + b - c = 2 \) holds true.

### Step 8: Solve for the side length
Let \( a = b = c \). Then substituting into \( 2ab - c^2 = 4 \):
\[ 2a^2 - a^2 = 4 \implies a^2 = 4 \implies a = 2 \]

### Step 9: Calculate the area
The area \( A \) of an equilateral triangle is given by:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substituting \( a = 2 \):
\[ A = \frac{\sqrt{3}}{4} \times 4 = \sqrt{3} \]

### Step 10: Find the square of the area
Thus, the square of the area is:
\[ A^2 = (\sqrt{3})^2 = 3 \]

### Final Answer
The square of the area of triangle ABC is **3**.

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