# Problem Description
This problem requires determining whether two given points should be connected to form a pentagon. The function takes a list of points and the indices of two points, and it returns whether these two points should be connected to form a valid pentagon. The points are not guaranteed to be in any particular order.

# Visual Facts
1. **Coordinate System Facts:**
   - The coordinate system is an 8x8 grid.
   - Points are positioned at integer coordinates within this grid.
   - All coordinates are positive and range from 1 to 8.

2. **Structure Facts:**
   - Each example contains exactly 5 points.
   - Each point is connected to exactly 2 other points.
   - The resulting shape is always a pentagon.
   - The pentagon is a closed figure with 5 sides.

3. **Layout Facts:**
   - The pentagon can be drawn in different orientations and scales.
   - The pentagon does not need to be symmetrical.
   - Points can be at different heights and distances from each other.

# Visual Patterns
1. **Connection Patterns:**
   - Points are connected in a sequence to form a closed loop.
   - Each point connects to its adjacent points in the sequence.
   - The sequence forms a pentagon without any intersections.

2. **Geometric Patterns:**
   - The five points form a convex pentagon when connected in sequence.
   - The pentagon is formed by connecting each point to its next neighbor in the sequence.
   - The sequence of points can be in clockwise or counter-clockwise order.

3. **Implementation Pattern:**
   - Points need to be sorted by their angular position relative to the center point.
   - Two points should be connected if they are adjacent in the sorted sequence.
   - The function should determine the correct sequence of points and check if the given points are adjacent in this 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) == 1 or (pos_b - pos_a) % len(points) == 1


# Test Script
def test():
    test_cases = [
        ([(2, 5), (4, 7), (3, 2), (5, 2), (6, 5)], 2, 4, False),
        ([(2, 5), (4, 7), (3, 2), (5, 2), (6, 5)], 2, 3, True),
        ([(2, 5), (4, 7), (3, 2), (5, 2), (6, 5)], 0, 4, False),
        ([(2, 5), (4, 7), (3, 2), (5, 2), (6, 5)], 0, 2, True),
        ([(2, 5), (4, 7), (3, 2), (5, 2), (6, 5)], 1, 3, False),
        ([(3, 8), (3, 0), (1, 1), (5, 6), (8, 2)], 0, 4, False),
        ([(3, 8), (3, 0), (1, 1), (5, 6), (8, 2)], 0, 1, False),
        ([(3, 8), (3, 0), (1, 1), (5, 6), (8, 2)], 1, 3, False),
        ([(3, 8), (3, 0), (1, 1), (5, 6), (8, 2)], 2, 4, False),
        ([(3, 8), (3, 0), (1, 1), (5, 6), (8, 2)], 2, 3, False),
    ]
    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()