If AB, BC and AC be the three sides of a triangle ABC, then which one of the following is true?

Options

A. `AB-BC=AC`
B. `(AB-BC)gtAC`
C. `(AB-BA) ltAC`
D. `AB^(2)-BC^(2)=AC^(2)`

Answer

To solve the problem regarding the sides of triangle ABC, we need to analyze the given options based on the properties of triangles.

### Step-by-Step Solution:

1. **Understanding Triangle Inequality Theorem**:
The triangle inequality theorem states that for any triangle with sides \( AB \), \( BC \), and \( AC \):
- \( AB + BC > AC \)
- \( AB + AC > BC \)
- \( AC + BC > AB \)

2. **Analyzing the Options**:
We have four options to evaluate:
- **Option 1**: \( AB - BC = AC \)
- **Option 2**: \( AB - BC > AC \)
- **Option 3**: \( AB - BA < AC \)
- **Option 4**: \( AB^2 - BC^2 = AC^2 \)

3. **Evaluating Each Option**:
- **Option 1**: \( AB - BC = AC \)
- This implies \( AB = AC + BC \). This cannot be true as it contradicts the triangle inequality (the sum of two sides must be greater than the third side).

- **Option 2**: \( AB - BC > AC \)
- Rearranging gives \( AB > AC + BC \), which again contradicts the triangle inequality.

- **Option 3**: \( AB - BA < AC \)
- This simplifies to \( AB < AC + BA \), which is valid as it is a rearrangement of the triangle inequality.

- **Option 4**: \( AB^2 - BC^2 = AC^2 \)
- This resembles the Pythagorean theorem, which is only true for right triangles, and cannot be generalized for all triangles.

4. **Conclusion**:
The only valid statement that holds true for any triangle is **Option 3: \( AB - BA < AC \)**. Thus, this is the correct answer.

### Final Answer:
The correct option is **Option 3: \( AB - BA < AC \)**.