# Problem Description
This is a geometric problem to determine whether two given points should be connected to form a valid five-pointed star (pentagram). The function needs to:
1. Take a list of 5 coordinate points and indices of 2 points
2. Determine if these two points should be connected based on the rules of pentagram construction
3. Return true if the points should be connected, false otherwise

# Visual Facts
1. Coordinate System Facts:
   - All examples use a standard 8x8 grid coordinate system
   - Points are positioned at integer coordinates
   - All coordinates are positive (1-8 range)

2. Structure Facts:
   - Each example contains exactly 5 points
   - Each point connects to exactly 2 other points
   - The resulting shape is always a five-pointed star
   - No intersections occur at points, only along line segments
   - The lines form 5 intersections within the pentagon

3. Layout Facts:
   - The star can be drawn in different orientations and scales
   - The star doesn't need to be symmetrical (as shown in Example 4)
   - Points can be at different heights and distances from each other

# Visual Patterns
1. Connection Patterns:
   - If we number points 1-5 in clockwise or counter-clockwise order:
     * Each point connects to points that are 2 steps away in the sequence
     * No adjacent points in the sequence are connected
     * The connection pattern follows: 1→3, 1→4, 2→4, 2→5, 3→5, 3→1, 4→1, 4→2, 5→2, 5→3

2. Geometric Patterns:
   - The five points form a convex pentagon when connected in sequence
   - The star is formed by connecting every second point in the sequence
   - There are exactly 5 intersection points formed by the line segments
   - Points must be arranged so that line segments can intersect to form the star pattern

3. Implementation Pattern:
   - Points need to be sorted by their angular position relative to the center point
   - The connection rule can be verified after establishing the correct sequence of points
   - Two points should be connected if their positions in the sorted sequence follow the connection pattern mentioned above

These patterns suggest that the solution will need to:
1. Find the center point of all points
2. Sort points by their angular position
3. Apply the connection rules based on the sorted sequence

# Solution
from math import atan2

def solution(points: list[tuple[int, int]], point_a_index: int, point_b_index: int) -> bool:
    """
    Given a list of points and the index of two points, determine whether the two points should be connected to form the shape in the figure.

    Args:
        points (list[tuple[int, int]]): An list of tuples, where each tuple consists of two integers representing the x and y coordinates of a point. The points are not guaranteed to be in any particular order.
        point_a_index (int): The 0-based index of the first point. 
        point_b_index (int): The 0-based index of the second point.

    Returns:
        bool: True if the two points should be connected, False otherwise.
    """
    centroid_x = sum((x for x, y in points)) / len(points)
    centroid_y = sum((y for x, y in points)) / len(points)

    def polar_angle(point):
        x, y = point
        return atan2(y - centroid_y, x - centroid_x)
    sorted_points = sorted(range(len(points)), key=lambda i: polar_angle(points[i]))
    pos_a = sorted_points.index(point_a_index)
    pos_b = sorted_points.index(point_b_index)
    return (pos_a - pos_b) % len(points) == 2 or (pos_b - pos_a) % len(points) == 2


# Test Script
def test():
    test_cases = [
        ([(2, 5), (4, 7), (3, 2), (5, 2), (6, 5)], 2, 4, True),
        ([(2, 5), (4, 7), (3, 2), (5, 2), (6, 5)], 2, 3, False),
        ([(2, 5), (4, 7), (3, 2), (5, 2), (6, 5)], 0, 4, True),
        ([(2, 5), (4, 7), (3, 2), (5, 2), (6, 5)], 0, 2, False),
        ([(2, 5), (4, 7), (3, 2), (5, 2), (6, 5)], 1, 3, True),
        ([(3, 8), (3, 0), (1, 1), (5, 6), (8, 2)], 0, 4, True),
        ([(3, 8), (3, 0), (1, 1), (5, 6), (8, 2)], 0, 1, True),
        ([(3, 8), (3, 0), (1, 1), (5, 6), (8, 2)], 1, 3, True),
        ([(3, 8), (3, 0), (1, 1), (5, 6), (8, 2)], 2, 4, True),
        ([(3, 8), (3, 0), (1, 1), (5, 6), (8, 2)], 2, 3, True),
    ]
    for points, point_a_index, point_b_index, expected_output in test_cases:
        try:
            output = solution(points, point_a_index, point_b_index)
        except Exception as e:
            error_message = (
                f"An exception occurred while running the test case:\n"
                f"`points`: {points}\n"
                f"`point_a_index`: {point_a_index}\n"
                f"`point_b_index`: {point_b_index}\n\n"
                f"Exception: '{str(e)}'"
            )
            assert False, error_message

        if output != expected_output:
            error_message = (
                f"The following Test Case failed:\n"
                f"`points`: {points}\n"
                f"`point_a_index`: {point_a_index}\n" 
                f"`point_b_index`: {point_b_index}\n\n"
                f"Actual Output: {output}\n"
                f"Expected Output: {expected_output}"
            )
            assert False, error_message

test()