from math import sqrt, pi, atan


def get_intersection(n: int) -> tuple[int, int]:
    """Compute the point where the line intersects the circle for a given
    number of circles `n`.

    Equation for circle: y = 1 - sqrt(2*x - x**2)
    Equation for line: y = (1/n) * x
    """

    # Computed via Wolfram alpha; could be derived by setting the curves equal,
    # isolating the root, and then applying the quadratic formula.
    x_int = (-sqrt(2) * n ** (3 / 2) + n**2 + n) / (n**2 + 1)
    y_int = x_int / n
    return (x_int, y_int)


def get_ratio_orange(n: int) -> float:
    """Compute the ratio between the orange and the blue section. The blue
    section is the L-section meaning the area outside of the quarter circle.
    The orange area is the concave triangle created by the diagonal line."""

    x, y = get_intersection(n)
    area_blue = 1 - (pi / 4)

    theta = atan((1 - x) / (1 - y))  # angle of concave triangle circle part
    area_sector = theta / 2  # area of sector of concave triangle circle part

    # area_orange = area triangle + area_rectangle + area_triangle - area_sector
    area_orange = (0.5 * x * y) + ((1 - x) * y) + (0.5 * (1 - x) * (1 - y))
    area_orange -= area_sector
    return area_orange / area_blue


def get_less_than(percentage: float) -> int:
    ratio = percentage / 100
    n_min, n_max = 1, 10000

    while n_max - n_min > 1:
        n_mid = (n_max + n_min) // 2
        ratio_n = get_ratio_orange(n_mid)
        if ratio_n < ratio:
            n_max = n_mid
        else:
            n_min = n_mid

    return n_max


def euler_587():
    assert get_ratio_orange(1) == 0.5
    assert abs(get_ratio_orange(2) - 0.3646) <= 0.0001
    assert get_less_than(10) == 15
    return get_less_than(0.1)


if __name__ == "__main__":
    solution = euler_587()
    print(f"e587.py: {solution}")
    assert solution == 2240