勾股四边形

给定四点，检查它们是否形成勾股四边形。
它被定义为整数 a, b, c, d 的元组，使得$a^2 + b^2 + c^2 = d^2$ .它们基本上是丢番图方程的解。在几何解释中，它表示一个边长为整数的长方体|a|, |b|, |c|其空间对角线为|d| .

此处显示的长方体边是勾股四边形的示例。
当它们的最大公约数为 1 时，它是原始的。每个勾股四元组都是原始四元组的整数倍。我们可以生成一组原始毕达哥拉斯四元组，其中 a 是奇数可以通过以下公式生成：

a = m² + n² – p² – q², 
b = 2(mq + np), 
c = 2(nq – mp), 
d = m² + n² + p² + q²

其中 m, n, p, q 是最大公约数为 1 的非负整数，使得 m + n + p + q 是奇数。因此，所有原始毕达哥拉斯四元组都以勒贝格恒等式为特征。

(m² + n² + p² + q²)² = (2mq + 2nq)² + 2(nq – mp)² + (m² + n² – p² – q²)m² + n² – p² – q²

C++

// C++ code to detect Pythagorean Quadruples.
#include 
using namespace std;
 
// function for checking
bool pythagorean_quadruple(int a, int b, int c,
                                        int d)
{
    int sum = a * a + b * b + c * c;
    if (d * d == sum)
        return true;
    else
        return false;
}
 
// Driver Code
int main()
{
    int a = 1, b = 2, c = 2, d = 3;
    if (pythagorean_quadruple(a, b, c, d))
        cout << "Yes" << endl;
    else
        cout << "No" << endl;
}

Java

// Java code to detect Pythagorean Quadruples.
import java.io.*;
import java.util.*;
 
class GFG {
 
// function for checking
static Boolean pythagorean_quadruple(int a, int b,
                                   int c, int d)
{
    int sum = a * a + b * b + c * c;
    if (d * d == sum)
        return true;
    else
        return false;
}
 
// Driver function
    public static void main (String[] args) {
    int a = 1, b = 2, c = 2, d = 3;
    if (pythagorean_quadruple(a, b, c, d))
        System.out.println("Yes");
    else
        System.out.println("No" );
         
    }
}
// This code is contributed by Gitanjali.

Python3

# Python  code to detect
# Pythagorean Quadruples.
import math
 
# function for checking
def pythagorean_quadruple(a,b, c, d):
 
    sum = a * a + b * b + c * c;
    if (d * d == sum):
        return True
    else:
        return False
 
#driver code
a = 1
b = 2
c = 2
d = 3
if (pythagorean_quadruple(a, b, c, d)):
    print("Yes")
else:
     print("No" )
 
# This code is contributed
# by Gitanjali.

C#

// C# code to detect
// Pythagorean Quadruples.
using System;
 
class GFG {
 
    // function for checking
    static Boolean pythagorean_quadruple(int a,
                            int b, int c, int d)
    {
        int sum = a * a + b * b + c * c;
        if (d * d == sum)
            return true;
        else
            return false;
    }
     
    // Driver function
        public static void Main () {
             
        int a = 1, b = 2, c = 2, d = 3;
         
        if (pythagorean_quadruple(a, b, c, d))
            Console.WriteLine("Yes");
        else
            Console.WriteLine("No" );
             
    }
}
 
// This code is contributed by vt_M.

Javascript

 
// JavaScript program to detect Pythagorean Quadruples.
 
// function for checking
function pythagorean_quadruple(a, b,
                                   c, d)
{
    let sum = a * a + b * b + c * c;
    if (d * d == sum)
        return true;
    else
        return false;
}
       
 
// Driver code
         
        let a = 1, b = 2, c = 2, d = 3;
    if (pythagorean_quadruple(a, b, c, d))
        document.write("Yes");
    else
        document.write("No" );
 

输出：

Yes

参考
维基
数学世界

如果您希望与专家一起参加现场课程，请参阅DSA 现场工作专业课程和学生竞争性编程现场课程。