If D is such a point on the side, BC of `DeltaABC` that `(AB)/(AC)=(BD)/(CD)`, then AD must be a/an:

Options

A. altitude of `DeltaABC`
B. median of `DeltaABC`
C. angle bisector of `DeltaABC`
D. perpendicular bisector of `DeltaABC`

Answer

To solve the problem, we need to analyze the given condition and determine what type of line segment AD represents in triangle ABC.

### Step-by-Step Solution:

1. **Understand the Given Condition**:
We are given that \( \frac{AB}{AC} = \frac{BD}{CD} \). This indicates a specific relationship between the segments of the triangle.

**Hint**: Look for properties of triangles that relate the sides and segments created by points on the sides.

2. **Identify the Point D**:
Point D is located on side BC of triangle ABC. We need to understand how this point relates to the sides AB and AC.

**Hint**: Consider how points on the sides of a triangle can affect the ratios of the triangle's sides.

3. **Recall the Angle Bisector Theorem**:
The Angle Bisector Theorem states that if a point D lies on the opposite side of a triangle from a vertex A, and if AD is the angle bisector of angle A, then the ratio of the lengths of the two segments created on the opposite side (BD and CD) is equal to the ratio of the other two sides (AB and AC).

**Hint**: Think about the definitions and properties of angle bisectors in triangles.

4. **Apply the Theorem to Our Case**:
Since we have \( \frac{AB}{AC} = \frac{BD}{CD} \), this directly implies that AD must be the angle bisector of angle A in triangle ABC.

**Hint**: Confirm that the condition given matches the criteria for the angle bisector theorem.

5. **Conclusion**:
Therefore, based on the analysis and application of the Angle Bisector Theorem, we conclude that AD is the angle bisector of triangle ABC.

**Final Answer**: AD must be an angle bisector of triangle ABC.