Given four points, check whether they form Pythagorean Quadruple.
It is defined as a tuple of integers a, b, c, d such that . They are basically the solutions of Diophantine Equations. In the geometric interpretation it represents a cuboid with integer side lengths |a|, |b|, |c| and whose space diagonal is |d| . The cuboids sides shown here are examples of pythagorean quadruples.
It is primitive when their greatest common divisor is 1. Every Pythagorean quadruple is an integer multiple of a primitive quadruple. We can generate the set of primitive pythagorean quadruples for which a is odd can be generated by formula :

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

where m, n, p, q are non-negative integers with greatest common divisor 1 such that m + n + p + q are odd. Thus, all primitive Pythagorean quadruples are characterized by Lebesgue’s identity.

(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.

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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.



Output:

Yes

References
Wiki
mathworld

tags:

Geometric Geometric