Proof:

Given: In triangle ABC, AD is the angle bisector of exterior angle A, and BD bisects angle B. Also, AB = AC.

To prove:
1) AD || BC
2) AB = AD

Proof of AD || BC:
We are given that AD is the angle bisector of exterior angle A. This means that angle BAD is equal to angle CAD.

Since BD bisects angle B, we can say that angle ABD is equal to angle CBD.

Now, let's assume that AD is not parallel to BC. This would mean that angle BAD and angle ABD are not equal, which contradicts the fact that AD is the angle bisector of exterior angle A.

Therefore, our assumption is wrong, and AD must be parallel to BC.

Proof of AB = AD:
To prove that AB = AD, we will use the given information that AB = AC.

Since AD is the angle bisector of exterior angle A, we can say that angle BAD is equal to angle CAD. And since AB = AC, we can conclude that triangle ABD is congruent to triangle ACD by the SAS (Side-Angle-Side) congruence criterion.

By congruence, we can say that angle ABD is equal to angle ACD.

Now, let's consider triangle ABD. We know that angle ABD is equal to angle ACD, and angle BAD is equal to angle CAD.

Using the Angle-Side-Angle (ASA) congruence criterion, we can conclude that triangle ABD is congruent to triangle ACD.

Since the corresponding sides of congruent triangles are equal, we can say that AB = AD.

Hence, we have proved that AB = AD.

Conclusion:
1) We have proved that AD is parallel to BC.
2) We have proved that AB = AD.

Hence, the statements are true based on the given information and the proofs provided.