Type of Triangle

You are given a 0-indexed integer array nums of size 3 which can form the sides of a triangle.

• A triangle is called equilateral if it has all sides of equal length.
• A triangle is called isosceles if it has exactly two sides of equal length.
• A triangle is called scalene if all its sides are of different lengths.

Return a string representing the type of triangle that can be formed or "none" if it cannot form a triangle.*

For example:

• nums = [3,3,3] should return "equilateral" because all sides are equal.
• nums = [3,4,5] should return "scalene" because no sides are equal and 3 + 4 > 5, 3 + 5 > 4, and 4 + 5 > 3.
• nums = [1,2,5] should return "none" because 1 + 2 is not greater than 5, meaning a triangle cannot be formed.

def triangleType(nums):
    """
    Given a 0-indexed integer array nums of size 3 which can form the sides of a triangle.

    A triangle is called equilateral if it has all sides of equal length.
    A triangle is called isosceles if it has exactly two sides of equal length.
    A triangle is called scalene if all its sides are of different lengths.

    Return a string representing the type of triangle that can be formed or "none" if it cannot form a triangle.
    """
    a, b, c = nums

    # Check if it can form a triangle
    if a + b <= c or a + c <= b or b + c <= a:
        return "none"

    # Check for equilateral
    if a == b == c:
        return "equilateral"

    # Check for isosceles
    if a == b or a == c or b == c:
        return "isosceles"

    # Otherwise, it's scalene
    return "scalene"

# Example Usage
nums1 = [3, 3, 3]
print(triangleType(nums1))  # Output: equilateral

nums2 = [3, 4, 5]
print(triangleType(nums2))  # Output: scalene

nums3 = [5, 4, 5]
print(triangleType(nums3))  # Output: isosceles

nums4 = [1, 2, 5]
print(triangleType(nums4))  # Output: none

Explanation:

1. Triangle Inequality Theorem: The most crucial part is to check if the given sides can form a triangle. According to the Triangle Inequality Theorem, the sum of the lengths of any two sides must be greater than the length of the third side. If this condition is not met, the sides cannot form a triangle, and the function should return "none".
2. Equilateral Triangle: If all three sides are equal (a == b == c), it's an equilateral triangle, and the function should return "equilateral".
3. Isosceles Triangle: If exactly two sides are equal (a == b or a == c or b == c), it's an isosceles triangle, and the function should return "isosceles".
4. Scalene Triangle: If all sides are different and the triangle inequality condition is met, it's a scalene triangle, and the function should return "scalene".

Big(O) Runtime Analysis:

The function performs a fixed number of comparisons and assignments, regardless of the input values. Therefore, the time complexity is O(1), which means it has constant time complexity.

Big(O) Space Usage Analysis:

The function uses a fixed amount of extra space to store the variables a, b, and c. The space used does not depend on the input values. Therefore, the space complexity is O(1), which means it has constant space complexity.

Edge Cases:

1. Zero or Negative Lengths: The problem statement specifies that 1 <= nums[i] <= 100. If the input constraints allowed zero or negative lengths, additional checks would be needed at the beginning of the function to ensure all sides are positive.
2. Large Numbers: If the numbers were very large, there could be a potential for integer overflow when checking the triangle inequality. This is not a concern here, as the constraint 1 <= nums[i] <= 100 ensures that the sums will not overflow.
3. Floating-Point Numbers: If the input consisted of floating-point numbers, you might need to consider the precision issues and use a small tolerance value for comparisons (e.g., abs(a - b) < tolerance). However, this problem deals only with integers.