If in a triangle ABC,` b cos ^(2) "" A/2 + a cos ^(2) "" B/2 = (3c)/(2),` then

Options

A. `c ^(2) ge ab`
B. `2c gt sqrt (ab)`
C. `(a+c)/(2c -a) + (b+c)/( 2c -b)` has minimum value 4
D. `a/c + c/b + b/a ge 3`

Answer

To solve the problem, we start with the given equation:

\[ b \cos^2 \left( \frac{A}{2} \right) + a \cos^2 \left( \frac{B}{2} \right) = \frac{3c}{2} \]

### Step 1: Use the identity for \(\cos^2\) of half angles
We know that:
\[ \cos^2 \left( \frac{A}{2} \right) = \frac{1 + \cos A}{2} \]
\[ \cos^2 \left( \frac{B}{2} \right) = \frac{1 + \cos B}{2} \]

### Step 2: Substitute the identities into the equation
Substituting these identities into the equation gives:
\[ b \left( \frac{1 + \cos A}{2} \right) + a \left( \frac{1 + \cos B}{2} \right) = \frac{3c}{2} \]

### Step 3: Simplify the equation
Multiply through by 2 to eliminate the fractions:
\[ b(1 + \cos A) + a(1 + \cos B) = 3c \]
This simplifies to:
\[ b + a + b \cos A + a \cos B = 3c \]

### Step 4: Rearranging the terms
Rearranging the equation gives:
\[ b \cos A + a \cos B = 3c - (a + b) \]

### Step 5: Use the cosine rule
Using the cosine rule in triangle \(ABC\):
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
\[ \cos B = \frac{a^2 + c^2 - b^2}{2ac} \]

### Step 6: Substitute \(\cos A\) and \(\cos B\) into the equation
Substituting these values into the equation:
\[ b \left( \frac{b^2 + c^2 - a^2}{2bc} \right) + a \left( \frac{a^2 + c^2 - b^2}{2ac} \right) = 3c - (a + b) \]

### Step 7: Simplify further
This results in:
\[ \frac{b(b^2 + c^2 - a^2)}{2c} + \frac{a(a^2 + c^2 - b^2)}{2c} = 3c - (a + b) \]

### Step 8: Combine the fractions
Combining the fractions on the left side:
\[ \frac{b(b^2 + c^2 - a^2) + a(a^2 + c^2 - b^2)}{2c} = 3c - (a + b) \]

### Step 9: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ b(b^2 + c^2 - a^2) + a(a^2 + c^2 - b^2) = 2c(3c - (a + b)) \]

### Step 10: Rearranging and solving for relationships
This equation can be rearranged to find relationships between \(a\), \(b\), and \(c\).

### Conclusion
After analyzing the relationships and conditions, we can conclude that the sides \(a\), \(b\), and \(c\) are in Arithmetic Progression (AP).