Given:
PA is a tangent to the circumcircle of triangle ABC.
PA is parallel to BC.

To prove:
AB = AC

Proof:

Step 1: Understanding the given information

- A tangent to a circle is a line that intersects the circle at exactly one point.
- In this case, PA is a tangent to the circumcircle of triangle ABC, which means it intersects the circle at point A.
- PA is also parallel to BC, which implies that PA and BC never intersect.

Step 2: Understanding the properties of tangents

- When a tangent intersects a circle, the angle formed between the tangent and the radius drawn to the point of tangency is always a right angle.
- In this case, the point of tangency is A, so the angle formed between the tangent PA and the radius drawn to point A is 90 degrees.

Step 3: Analyzing the given situation

- Since PA is parallel to BC, and the angle formed between PA and the radius drawn to point A is 90 degrees, the angle formed between BC and the radius drawn to point A must also be 90 degrees.
- This implies that triangle ABC is a right-angled triangle with the right angle at point A.

Step 4: Applying the properties of right-angled triangles

- In a right-angled triangle, the sides opposite the acute angles are called the legs, and the side opposite the right angle is called the hypotenuse.
- In triangle ABC, AB and AC are the legs, and BC is the hypotenuse.
- According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
- Since PA is parallel to BC, AB and AC are the corresponding sides of the right-angled triangle formed by PA, AB, and AC.
- Therefore, AB^2 + AC^2 = BC^2.

Step 5: Using the given information to prove AB = AC

- We know that PA is parallel to BC, which means that AB and AC are corresponding sides of the right-angled triangle formed by PA, AB, and AC.
- Since the corresponding sides of a right-angled triangle are equal, we can conclude that AB = AC.

Conclusion:
Therefore, by analyzing the given information and using the properties of tangents and right-angled triangles, we have proven that if PA is a tangent to the circumcircle of triangle ABC and PA is parallel to BC, then AB = AC.