CF is the internal bisector of angle C of `Delta ABC` then CF is equal to

Options

A. `(2ab)/(a+b) cos"C/2`
B. `(a+b)/(2ab) cos " C/2`
C. `(b sin A)/(sin(B+C/2))`
D. none of these

Answer

To find the length of the internal bisector CF of angle C in triangle ABC, we can use the angle bisector theorem and some properties of triangles. Here’s a step-by-step solution:

### Step 1: Understand the Triangle and the Angle Bisector
We have triangle ABC with angle C being bisected by line segment CF. According to the angle bisector theorem, the ratio of the lengths of the two segments created by the bisector on the opposite side (AB) is equal to the ratio of the other two sides of the triangle.

### Step 2: Apply the Angle Bisector Theorem
According to the angle bisector theorem:
\[ \frac{AF}{FB} = \frac{AC}{BC} \]
Let \( AF = m \) and \( FB = n \). Therefore, we can write:
\[ \frac{m}{n} = \frac{AC}{BC} \]

### Step 3: Express the Length of CF
The length of the angle bisector CF can be calculated using the formula:
\[ CF = \frac{2 \cdot AB \cdot AC \cdot BC}{(AC + BC) \cdot \cos\left(\frac{C}{2}\right)} \]
Where \( AB \) is the length of side opposite to angle C, \( AC \) and \( BC \) are the lengths of the other two sides.

### Step 4: Substitute Values
If we denote:
- \( a = BC \)
- \( b = AC \)
- \( c = AB \)

Then we can rewrite the length of CF as:
\[ CF = \frac{2ab}{a + b} \cdot \cos\left(\frac{C}{2}\right) \]

### Step 5: Final Expression
Thus, the final expression for the length of the internal bisector CF is:
\[ CF = \frac{2ab}{a + b} \cdot \cos\left(\frac{C}{2}\right) \]

### Summary
The length of the internal bisector CF of angle C in triangle ABC can be calculated using the formula:
\[ CF = \frac{2ab}{a + b} \cdot \cos\left(\frac{C}{2}\right) \]