Problem Statement:
PQR is an equilateral triangle inscribed in a circle A, B are points on the arcs PQ, QR such that PA = QB. Prove that AP = AQ = PB.

Proof:
To prove that AP = AQ = PB, we will use the properties of an equilateral triangle and the fact that PQR is inscribed in a circle.

1. Draw the Diagram:
First, let's draw a diagram to visualize the problem. We have an equilateral triangle PQR inscribed in a circle. Points A and B are on the arcs PQ and QR respectively, such that PA = QB.

2. Inscribed Angles:
Since PQR is an equilateral triangle, each angle of the triangle is 60 degrees. When a triangle is inscribed in a circle, the measure of an inscribed angle is equal to half the measure of its intercepted arc.

- Angle PAQ = 60/2 = 30 degrees. (Arc PQ)
- Angle PBA = 60/2 = 30 degrees. (Arc PQ)

3. Central Angles:
The measure of a central angle is equal to the measure of its intercepted arc.

- Angle PAQ = Angle PBQ. (Arc PQ)

4. Congruent Angles:
Using the properties of an equilateral triangle, we know that all angles of the triangle are equal.

- Angle PAQ = Angle APQ = Angle AQ = 60 degrees.
- Angle PBQ = Angle PQR = Angle PQ = 60 degrees.

5. Triangle Congruence:
We can conclude that triangles PAQ and PBQ are congruent by angle-angle-side (AAS) congruence.

- Angle PAQ = Angle PBQ (proved in step 3)
- Angle APQ = Angle PQR (proved in step 4)
- Side AQ = Side BQ (given)

6. Corresponding Parts of Congruent Triangles:
In congruent triangles, corresponding parts are equal.

- AP = BP (corresponding sides of congruent triangles PAQ and PBQ)
- AQ = BQ (corresponding sides of congruent triangles PAQ and PBQ)

7. Final Conclusion:
From step 6, we can conclude that AP = AQ = PB. Thus, the statement is proved.