Angle bisector of angles subtended by the chord of a circle in the same segmetn DOES NOT represent

Options

A. A family of lines passing through a fixed point lying on the circle
B. A family of lines passing through a fixed point inside the circle
C. A family of lines passing through a fixed point lying outside the circle
D. A family of lines through which never pass through a fixed point

Explanation

To solve the question regarding the angle bisector of angles subtended by a chord of a circle in the same segment, we will analyze the properties of angle bisectors and their relation to the circle.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let \( A \) and \( B \) be the endpoints of a chord in a circle.
- Let \( P \) be any point on the arc \( AB \) (not containing the chord).
- The angles \( \angle A P B \) is subtended by the chord \( AB \) at point \( P \).

**Hint**: Visualize the circle and the chord \( AB \) along with the point \( P \) on the arc.

2. **Angle Bisectors**:
- The angle bisector of \( \angle A P B \) will divide this angle into two equal angles.
- If we draw angle bisectors from points \( P_1, P_2, P_3 \) on the arc \( AB \), they will all intersect at a single point \( P \) inside the circle.

**Hint**: Remember that angle bisectors of the same angle will converge at a point.

3. **Properties of the Angle Bisectors**:
- All angle bisectors from points on the arc \( AB \) will pass through a fixed point \( P \) which lies on the circle.
- This means that the angle bisectors do not represent lines that pass through points outside the circle or points inside the circle that are not on the arc.

**Hint**: Consider where the angle bisectors can intersect based on their definitions.

4. **Analyzing the Options**:
- **Option A**: A family of lines passing through a fixed point lying on the circle. (Correct)
- **Option B**: A family of lines passing through a fixed point inside the circle. (Incorrect)
- **Option C**: A family of lines passing through a fixed point lying outside the circle. (Correct)
- **Option D**: A family of lines which never pass through a fixed point. (Correct)

**Hint**: Identify which options align with the properties of the angle bisectors derived from the chord.

5. **Conclusion**:
- The only option that does not represent the angle bisector of angles subtended by the chord of a circle in the same segment is **Option B**, which states that they pass through a fixed point inside the circle.

### Final Answer:
The angle bisector of angles subtended by the chord of a circle in the same segment does not represent **Option B: A family of lines passing through a fixed point inside the circle**.