In which of the following situations, it is possible to have `a DeltaABC` ?
(All symbols used have usual meaning in a triangle)

Options

A. (a)` (a+c-b) (a-c+b) =4bc`
B. (b)`b^(2)sin 2C+cos ^(2)sin2B=ab`
C. (c)`a=3,b=5, c=7 and C =(2pi)/(3)`
D. (d)`cos ((A-C)/(2))=cos ((A+C)/(2))`

Answer

To determine in which of the given situations it is possible to have a triangle \( \Delta ABC \), we will analyze each option based on the properties of triangles.

### Step-by-Step Solution:

1. **Understanding Triangle Properties**:
A triangle can be formed if it satisfies certain conditions, such as the triangle inequality theorem and specific properties of angles and sides.

2. **Analyzing Option A**:
The condition given is \( a + b > c \), \( a + c > b \), and \( b + c > a \). We need to check if this condition holds true for the given values.
- If the values do not satisfy these inequalities, then a triangle cannot be formed.

3. **Analyzing Option B**:
The condition is \( B^2 \sin(2C) + O^2 \sin(2B) = AB \).
- This is a valid property for triangles. If this equation holds true, a triangle can be formed.

4. **Analyzing Option C**:
The values given are \( a = 3 \), \( b = 5 \), and \( c = 7 \) with \( C = \frac{2\pi}{3} \).
- We can check the angle sum property: \( A + B + C = \pi \). If this holds true, a triangle can be formed.

5. **Analyzing Option D**:
The condition is \( \cos A + \frac{b}{2} = \cos B \).
- This is another valid property for triangles. If this equation holds true, a triangle can be formed.

6. **Conclusion**:
After analyzing all options, we find that options B, C, and D satisfy the conditions for forming a triangle, while option A does not meet the necessary conditions.

### Final Answer:
- The possible situations to have \( \Delta ABC \) are **B, C, and D**.