The next step in constructing a regular hexagon follows the standard method of using a compass and straightedge. Given the steps already performed, the best next step would be:

"With compass setting the same as line segment PA, place the compass at C and draw an arc that intersects the circle."

Explanation:

1. A regular hexagon is constructed by marking six equal segments along the circumference of a circle.
2. You started with a circle centered at P and marked points A, B, and C along the edge.
3. By connecting A to B and A to C, you began forming the initial hexagon framework.
4. The standard method involves stepping off equal-length segments (equal to the circle's radius) around the circle using the compass.
5. Since the given step instructs placing the compass at C with the same radius PA, it ensures the continuation of marking the hexagon’s vertices.

Would you like further clarification or a step-by-step breakdown of hexagon construction?

Related Questions:

1. How do you construct a regular hexagon using only a compass and straightedge?
2. Why does a hexagon fit perfectly inside a circle when using this method?
3. What are some practical applications of hexagonal tiling in nature and engineering?
4. How can you verify that all sides of the constructed hexagon are equal?
5. How do you construct an inscribed equilateral triangle using the same method?

Tip:

A regular hexagon can be thought of as six equilateral triangles joined together, which is why using the radius of the circle to mark points along the circumference works perfectly!