To show that $AD$ is the perpendicular bisector of side $BC$ in $\triangle ABC$, given that $AB = AC$ and $AD$ bisects $\angle BAC$, let's go through the necessary steps and use some properties of isosceles triangles and angle bisectors.

Step-by-Step Solution:

1. Identify Known Values and Properties:

   • Given $\triangle ABC$ where $AB = AC$, we know that $\triangle ABC$ is isosceles.
   • $AD$ is the angle bisector of $\angle BAC$.

2. Properties of the Angle Bisector in an Isosceles Triangle:

   • In an isosceles triangle, the angle bisector of the vertex angle (in this case, $\angle BAC$) is also the perpendicular bisector of the opposite side. This property is specific to isosceles triangles.

3. Implications of the Angle Bisector:

   • Since $AD$ bisects $\angle BAC$, it implies that $\triangle ABD$ and $\triangle ACD$ are congruent by the Angle-Side-Angle (ASA) criterion (because $AB = AC$, $\angle BAD = \angle CAD$, and $AD$ is common to both triangles).

4. Conclusion:

   • By congruence, $BD = DC$ (corresponding parts of congruent triangles).
   • Additionally, because $\triangle ABC$ is isosceles and $AD$ is the angle bisector, $AD$ must be perpendicular to $BC$ at point $D$.

Therefore, $AD$ is both the perpendicular bisector of $BC$ and the angle bisector of $\angle BAC$.

Would you like more details or have any questions?

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Here are five related questions for further exploration:

1. What is the general rule for the angle bisector of an isosceles triangle?
2. How can we prove that two triangles are congruent using ASA criterion?
3. Can the perpendicular bisector of a side always be considered an angle bisector in any triangle?
4. How does the property of perpendicular bisectors apply to equilateral triangles?
5. What other properties emerge when an angle bisector is also a median in an isosceles triangle?

Tip: In isosceles triangles, remember that the angle bisector of the vertex angle is always perpendicular to the base and bisects it.