Given:
- Triangle ABC is an equilateral triangle.
- Line AP intersects side BC at point P.
- Line AP intersects the circumcircle of triangle ABC at point Q.
- BQ = 4 cm
- CQ = 3 cm

To find:
The length of PQ.

Explanation:

1. Properties of an Equilateral Triangle:
- In an equilateral triangle, all sides are equal in length.
- The angles of an equilateral triangle are all 60 degrees.

2. Key Point: Angle BAC:
- Let's consider angle BAC of triangle ABC.
- Since triangle ABC is an equilateral triangle, all angles are 60 degrees.
- So, angle BAC = 60 degrees.

3. Key Point: Angle BCQ:
- Angle BCQ is an inscribed angle that intercepts arc BAC on the circumcircle of triangle ABC.
- According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of its intercepted arc.
- So, angle BCQ = 60/2 = 30 degrees.

4. Key Point: Angle BQC:
- Angle BQC is an exterior angle of triangle BCQ.
- The measure of an exterior angle of a triangle is equal to the sum of the measures of its interior opposite angles.
- So, angle BQC = angle BCQ + angle QCB = 30 + 60 = 90 degrees.

5. Key Point: Right Triangle BQC:
- Triangle BQC is a right triangle with angle BQC = 90 degrees.
- BQ = 4 cm and CQ = 3 cm are the lengths of the legs of the right triangle.

6. Applying Pythagorean Theorem:
- Using the Pythagorean Theorem, we can find the length of the hypotenuse BQ.
- The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
- So, BQ^2 = BQ^2 + CQ^2.
- Substituting the given values, we get BQ^2 = 4^2 + 3^2 = 16 + 9 = 25.
- Taking the square root of both sides, we find BQ = 5 cm.

7. Key Point: Triangle APQ:
- Triangle APQ is a similar triangle to triangle BCQ.
- This is because angle BCQ and angle APQ are corresponding angles formed by line AP intersecting the same arc BAC on the circumcircle.
- Since angle BCQ = angle APQ = 30 degrees, the two triangles are similar.

8. Applying Proportions:
- Since triangle APQ is similar to triangle BCQ, we can set up a proportion to find the length of PQ.
- BQ/PQ = CQ/AQ.
- Substituting the given values, we have 5/PQ = 3/AQ.
- Rearranging the equation, we get PQ = (5 * AQ) / 3.

9. Finding AQ:
- To find AQ, we can use