检查直角三角形是否对大边有效

给定三个整数a，b和c作为三元组。检查是否可以制作直角三角形。如果可能的话，否则没有打印是。 10 ^-18 <= a，b，c <= 10 ¹⁸

例子：

Input: 3 4 5
Output: Yes
Explanation:
Since 3*3 + 4*4 = 5*5
Hence print "Yes"

Input: 8 5 13
Since 8 + 5 < 13 which violates the property of
triangle. Hence print "No"

为了使直角三角形有效，它必须满足以下条件：

1. a，b和c应该大于0。
2. 三角形的任意两个边的和必须大于第三个边。
3. 勾股定理，即a ² + b ² = c ² 。

前两个条件很容易检查，但第三个条件我们必须注意溢出。由于a，b和c可能很大，因此除非我们在Java使用Python或BigInteger库，否则我们无法直接比较它们。对于C和C++之类的语言，我们必须减少分数形式的表达。
$\implies a^2 + b^2 = c^2 \implies a^2 = c^2 - b^2 \implies \dfrac{a}{c-b}=\dfrac{c+b}{a}$
在比较分数之前，我们需要通过将它们的分子和分母除以gcd来将它们转换为简化形式。现在比较LHS和RHS的两个分数的分子和分母，以便如果两者都相同，则表示有效的直角三角形，否则表示有效的直角三角形。

C++

// C++ program to check validity of triplets
#include 
using namespace std;
  
// Function to check pythagorean triplets
bool Triplets(long long a, long long b, long long c)
{
    if (a <= 0 || b <= 0 || c <= 0)
        return false;
  
    vector vec{ a, b, c };
    sort(vec.begin(), vec.end());
  
    // Re-initialize a, b, c in ascending order
    a = vec[0], b = vec[1], c = vec[2];
  
    // Check validation of sides of triangle
    if (a + b <= c)
        return false;
  
    long long p1 = a, p2 = c - b;
  
    // Reduce fraction to simplified form
    long long div = __gcd(p1, p2);
    p1 /= div, p2 /= div;
  
    long long q1 = c + b, q2 = a;
  
    // Reduce fraction to simplified form
    div = __gcd(q1, q2);
    q1 /= div, q2 /= div;
  
    // If fraction are equal return
    // 'true' else 'false'
    return (p1 == q1 && p2 == q2);
}
  
// Function that will return 'Yes' or 'No'
// according to the correction of triplets
string checkTriplet(long long a, long long b, long long c)
{
    if (Triplets(a, b, c))
        return "Yes";
    else
        return "No";
}
  
// Driver code
int main()
{
    long long a = 4, b = 3, c = 5;
    cout << checkTriplet(a, b, c) << endl;
  
    a = 8, b = 13, c = 5;
    cout << checkTriplet(a, b, c) << endl;
  
    a = 1200000000000, b = 1600000000000,
    c = 2000000000000;
    cout << checkTriplet(a, b, c) << endl;
  
    return 0;
}

Java

// Java program to check validity of triplets
import java.util.*;
  
class GFG 
{
      
// Function to check pythagorean triplets
static boolean Triplets(long a, 
                        long b, long c)
{
    if (a <= 0 || b <= 0 || c <= 0)
        return false;
  
    long []vec = { a, b, c };
    Arrays.sort(vec);
  
    // Re-initialize a, b, c in ascending order
    a = vec[0]; b = vec[1]; c = vec[2];
  
    // Check validation of sides of triangle
    if (a + b <= c)
        return false;
  
    long p1 = a, p2 = c - b;
  
    // Reduce fraction to simplified form
    long div = __gcd(p1, p2);
    p1 /= div; p2 /= div;
  
    long q1 = c + b, q2 = a;
  
    // Reduce fraction to simplified form
    div = __gcd(q1, q2);
    q1 /= div; q2 /= div;
  
    // If fraction are equal return
    // 'true' else 'false'
    return (p1 == q1 && p2 == q2);
}
  
// Function that will return 'Yes' or 'No'
// according to the correction of triplets
static String checkTriplet(long a, 
                           long b, long c)
{
    if (Triplets(a, b, c))
        return "Yes";
    else
        return "No";
}
  
static long __gcd(long a, long b) 
{ 
    if (b == 0) 
        return a; 
    return __gcd(b, a % b); 
      
}
  
// Driver code
public static void main(String[] args) 
{
    long a = 4, b = 3, c = 5;
    System.out.println(checkTriplet(a, b, c));
  
    a = 8; b = 13; c = 5;
    System.out.println(checkTriplet(a, b, c));
  
    a = 1200000000000L; b = 1600000000000L;
    c = 2000000000000L;
    System.out.println(checkTriplet(a, b, c));
}
}
  
// This code is contributed 
// by Princi Singh

Python3

# Python3 program to check validity of triplets 
def Triplets(a, b, c):
      
    if (a <= 0 or b <= 0 or c <= 0):
        return False
          
    vec = [ a, b, c ]
    vec.sort()
  
    # Re - initialize a, b, c in ascending order
    a = vec[0]; b = vec[1]; c = vec[2]
  
    # Check validation of sides of triangle
    if (a + b <= c):
        return False
  
    p1 = a; p2 = c - b
  
    # Reduce fraction to simplified form
    div = __gcd(p1, p2)
    p1 //= div
    p2 //= div
  
    q1 = c + b
    q2 = a
  
    # Reduce fraction to simplified form
    div = __gcd(q1, q2)
    q1 //= div
    q2 //= div
  
    # If fraction are equal return
    # 'true' else 'false'
    return (p1 == q1 and p2 == q2)
  
# Function that will return 'Yes' or 'No'
# according to the correction of triplets
def checkTriplet(a, b, c):
      
    if (Triplets(a, b, c)):
        return "Yes"
    else:
        return "No"
  
def __gcd(a, b):
    if (b == 0):
        return a
    return __gcd(b, a % b)
  
# Driver code
a = 4
b = 3
c = 5
print(checkTriplet(a, b, c))
  
a = 8
b = 13
c = 5
print(checkTriplet(a, b, c))
  
a = 1200000000000 
b = 1600000000000
c = 2000000000000
print(checkTriplet(a, b, c))
  
# This code is contributed by ng24_7

C#

// C# program to check validity of triplets
using System;
      
class GFG 
{
      
// Function to check pythagorean triplets
static Boolean Triplets(long a, 
                        long b, long c)
{
    if (a <= 0 || b <= 0 || c <= 0)
        return false;
  
    long []vec = { a, b, c };
    Array.Sort(vec);
  
    // Re-initialize a, b, c in ascending order
    a = vec[0]; b = vec[1]; c = vec[2];
  
    // Check validation of sides of triangle
    if (a + b <= c)
        return false;
  
    long p1 = a, p2 = c - b;
  
    // Reduce fraction to simplified form
    long div = __gcd(p1, p2);
    p1 /= div; p2 /= div;
  
    long q1 = c + b, q2 = a;
  
    // Reduce fraction to simplified form
    div = __gcd(q1, q2);
    q1 /= div; q2 /= div;
  
    // If fraction are equal return
    // 'true' else 'false'
    return (p1 == q1 && p2 == q2);
}
  
// Function that will return 'Yes' or 'No'
// according to the correction of triplets
static String checkTriplet(long a, 
                        long b, long c)
{
    if (Triplets(a, b, c))
        return "Yes";
    else
        return "No";
}
  
static long __gcd(long a, long b) 
{ 
    if (b == 0) 
        return a; 
    return __gcd(b, a % b); 
      
}
  
// Driver code
public static void Main(String[] args) 
{
    long a = 4, b = 3, c = 5;
    Console.WriteLine(checkTriplet(a, b, c));
  
    a = 8; b = 13; c = 5;
    Console.WriteLine(checkTriplet(a, b, c));
  
    a = 1200000000000L; b = 1600000000000L;
    c = 2000000000000L;
    Console.WriteLine(checkTriplet(a, b, c));
}
}
  
// This code has been contributed by 29AjayKumar

输出：

Yes
No
Yes

时间复杂度： O(log(M))，其中M是a，b和c中的最大值。
辅助空间： O(1)