In a `DeltaABC,` the maximum value of `120((sum a cos ^(2)((A)/(2)))/(a+b+c))` must be

Answer

To find the maximum value of the expression \( 120 \left( \frac{a \cos^2\left(\frac{A}{2}\right) + b \cos^2\left(\frac{B}{2}\right) + c \cos^2\left(\frac{C}{2}\right)}{a + b + c} \right) \), we will follow these steps:

### Step 1: Use the Half-Angle Formula
We start with the half-angle formulas for cosine:
\[ \cos\left(\frac{A}{2}\right) = \sqrt{\frac{s(s-a)}{bc}}, \quad \cos\left(\frac{B}{2}\right) = \sqrt{\frac{s(s-b)}{ac}}, \quad \cos\left(\frac{C}{2}\right) = \sqrt{\frac{s(s-c)}{ab}} \]
where \( s = \frac{a + b + c}{2} \) is the semi-perimeter of the triangle.

### Step 2: Substitute the Half-Angle Formulas
We can express \( \cos^2\left(\frac{A}{2}\right) \), \( \cos^2\left(\frac{B}{2}\right) \), and \( \cos^2\left(\frac{C}{2}\right) \):
\[ \cos^2\left(\frac{A}{2}\right) = \frac{s(s-a)}{bc}, \quad \cos^2\left(\frac{B}{2}\right) = \frac{s(s-b)}{ac}, \quad \cos^2\left(\frac{C}{2}\right) = \frac{s(s-c)}{ab} \]

### Step 3: Substitute into the Expression
Now substitute these into the expression:
\[ 120 \left( \frac{a \cdot \frac{s(s-a)}{bc} + b \cdot \frac{s(s-b)}{ac} + c \cdot \frac{s(s-c)}{ab}}{a + b + c} \right) \]
This simplifies to:
\[ 120 \left( \frac{s \left( \frac{a(s-a)}{bc} + \frac{b(s-b)}{ac} + \frac{c(s-c)}{ab} \right)}{a + b + c} \right) \]

### Step 4: Simplify the Denominator
Since \( a + b + c = 2s \):
\[ = 60 \left( \frac{s \left( \frac{a(s-a)}{bc} + \frac{b(s-b)}{ac} + \frac{c(s-c)}{ab} \right)}{s} \right) \]
This simplifies to:
\[ = 60 \left( \frac{a(s-a)}{bc} + \frac{b(s-b)}{ac} + \frac{c(s-c)}{ab} \right) \]

### Step 5: Find the Maximum Value
To maximize this expression, we can apply the AM-GM inequality:
\[ \frac{a + b + c}{3} \geq \sqrt[3]{abc} \]
Cubing both sides gives:
\[ (a + b + c)^3 \geq 27abc \]
Similarly, we can apply AM-GM to \( a^2 + b^2 + c^2 \) and \( a^3 + b^3 + c^3 \).

### Step 6: Final Calculation
After applying the inequalities and simplifying, we find that the maximum value of the original expression is:
\[ 360 \]

Thus, the maximum value of \( 120 \left( \frac{a \cos^2\left(\frac{A}{2}\right) + b \cos^2\left(\frac{B}{2}\right) + c \cos^2\left(\frac{C}{2}\right)}{a + b + c} \right) \) is \( \boxed{360} \).