# Program to check whether 4 points in a 3-D plane are Coplanar

Given 4 points (x1, y1, z1), (x2, y2, z2), (x3, y3, z3), (x4, y4, z4). The task is to write a program to check whether these 4 points are coplanar or not.

Note: 4 points in a 3-D plane are said to be coplanar if they lies in the same plane. Examples:

```Input:
x1 = 3, y1 = 2, z1 = -5
x2 = -1, y2 = 4, z2 = -3
x3 = -3, y3 = 8, z3 = -5
x4 = -3, y4 = 2, z4 = 1
Output: Coplanar

Input:
x1 = 0, y1 = -1, z1 = -1
x2 = 4, y2 = 5, z2 = 1
x3 = 3, y3 = 9, z3 = 4
x4 = -4, y4 = 4, z4 = 3
Output: Not Coplanar```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:

1. To check whether 4 points are coplanar or not, first of all, find the equation of the plane passing through any three of the given points.
Approach to find equation of a plane passing through 3 points.
2. Then, check whether the 4th point satisfies the equation obtained in step 1. That is, putting the value of 4th point in the equation obtained. If it satisfies the equation then the 4 points are Coplanar otherwise not.

Below is the implementation of the above idea:

## C++

 `// C++ program to check if 4 points  ` `// in a 3-D plane are Coplanar  ` ` `  `#include ` `using` `namespace` `std ; ` ` `  `// Function to find equation of plane. ` `void` `equation_plane(``int` `x1,``int` `y1,``int` `z1,``int` `x2,``int` `y2,``int` `z2, ` `             ``int` `x3, ``int` `y3, ``int` `z3, ``int` `x, ``int` `y, ``int` `z) ` `    ``{ ` `    ``int` `a1 = x2 - x1 ; ` `    ``int` `b1 = y2 - y1 ; ` `    ``int` `c1 = z2 - z1 ; ` `    ``int` `a2 = x3 - x1 ; ` `    ``int` `b2 = y3 - y1 ; ` `    ``int` `c2 = z3 - z1 ; ` `    ``int` `a = b1 * c2 - b2 * c1 ; ` `    ``int` `b = a2 * c1 - a1 * c2 ; ` `    ``int` `c = a1 * b2 - b1 * a2 ; ` `    ``int` `d = (- a * x1 - b * y1 - c * z1) ; ` `       `  `    ``// equation of plane is: a*x + b*y + c*z = 0 #  ` `       `  `    ``// checking if the 4th point satisfies  ` `    ``// the above equation  ` `    ``if``(a * x + b * y + c * z + d == 0)  ` `        ``cout << ``"Coplanar"` `<< endl;  ` `    ``else` `        ``cout << ``"Not Coplanar"` `<< endl;  ` `                  `  `    ``} ` `     `  `// Driver Code  ` `int` `main() ` `{ ` `      `  `int` `x1 = 3 ; ` `int` `y1 = 2 ; ` `int` `z1 = -5 ; ` `int` `x2 = -1 ; ` `int` `y2 = 4 ; ` `int` `z2 = -3 ; ` `int` `x3 = -3 ; ` `int` `y3 = 8 ; ` `int` `z3 = -5 ; ` `int` `x4 = -3 ; ` `int` `y4 = 2 ; ` `int` `z4 = 1 ; ` ` `  `// function calling ` `equation_plane(x1, y1, z1, x2, y2, z2, x3,   ` `                            ``y3, z3, x4, y4, z4) ;                             ` `return` `0; ` ` `  `// This code is contributed by ANKITRAI1 ` `} `

## Java

 `//Java program to check if 4 points  ` `//in a 3-D plane are Coplanar  ` ` `  `public` `class` `GFG { ` ` `  `    ``//Function to find equation of plane. ` `    ``static` `void` `equation_plane(``int` `x1,``int` `y1,``int` `z1,``int` `x2,``int` `y2,``int` `z2, ` `              ``int` `x3, ``int` `y3, ``int` `z3, ``int` `x, ``int` `y, ``int` `z) ` `     ``{ ` `     ``int` `a1 = x2 - x1 ; ` `     ``int` `b1 = y2 - y1 ; ` `     ``int` `c1 = z2 - z1 ; ` `     ``int` `a2 = x3 - x1 ; ` `     ``int` `b2 = y3 - y1 ; ` `     ``int` `c2 = z3 - z1 ; ` `     ``int` `a = b1 * c2 - b2 * c1 ; ` `     ``int` `b = a2 * c1 - a1 * c2 ; ` `     ``int` `c = a1 * b2 - b1 * a2 ; ` `     ``int` `d = (- a * x1 - b * y1 - c * z1) ; ` `         `  `     ``// equation of plane is: a*x + b*y + c*z = 0 #  ` `         `  `     ``// checking if the 4th point satisfies  ` `     ``// the above equation  ` `     ``if``(a * x + b * y + c * z + d == ``0``)  ` `         ``System.out.println(``"Coplanar"``);  ` `     ``else` `         ``System.out.println(``"Not Coplanar"``);  ` `                    `  `     ``} ` `       `  `    ``//Driver Code  ` `    ``public` `static` `void` `main(String[] args) { ` `         `  `        ``int` `x1 = ``3` `; ` `        ``int` `y1 = ``2` `; ` `        ``int` `z1 = -``5` `; ` `        ``int` `x2 = -``1` `; ` `        ``int` `y2 = ``4` `; ` `        ``int` `z2 = -``3` `; ` `        ``int` `x3 = -``3` `; ` `        ``int` `y3 = ``8` `; ` `        ``int` `z3 = -``5` `; ` `        ``int` `x4 = -``3` `; ` `        ``int` `y4 = ``2` `; ` `        ``int` `z4 = ``1` `; ` ` `  `        ``//function calling ` `        ``equation_plane(x1, y1, z1, x2, y2, z2, x3,   ` `                                 ``y3, z3, x4, y4, z4) ;                             ` `    ``} ` `} `

## Python3

 `# Python program to check if 4 points ` `# in a 3-D plane are Coplanar ` ` `  `# Function to find equation of plane. ` `def` `equation_plane(x1, y1, z1, x2, y2, z2, x3,  ` `                                ``y3, z3, x, y, z):  ` `     `  `    ``a1 ``=` `x2 ``-` `x1 ` `    ``b1 ``=` `y2 ``-` `y1 ` `    ``c1 ``=` `z2 ``-` `z1 ` `    ``a2 ``=` `x3 ``-` `x1 ` `    ``b2 ``=` `y3 ``-` `y1 ` `    ``c2 ``=` `z3 ``-` `z1 ` `    ``a ``=` `b1 ``*` `c2 ``-` `b2 ``*` `c1 ` `    ``b ``=` `a2 ``*` `c1 ``-` `a1 ``*` `c2 ` `    ``c ``=` `a1 ``*` `b2 ``-` `b1 ``*` `a2 ` `    ``d ``=` `(``-` `a ``*` `x1 ``-` `b ``*` `y1 ``-` `c ``*` `z1) ` `     `  `    ``# equation of plane is: a*x + b*y + c*z = 0 # ` `     `  `    ``# checking if the 4th point satisfies ` `    ``# the above equation ` `    ``if``(a ``*` `x ``+` `b ``*` `y ``+` `c ``*` `z ``+` `d ``=``=` `0``): ` `        ``print``(``"Coplanar"``) ` `    ``else``: ` `        ``print``(``"Not Coplanar"``) ` `     `  `     `  `# Driver Code  ` `x1 ``=` `3` `y1 ``=` `2` `z1 ``=` `-``5` `x2 ``=` `-``1` `y2 ``=` `4` `z2 ``=` `-``3` `x3 ``=` `-``3` `y3 ``=` `8` `z3 ``=` `-``5` `x4 ``=` `-``3` `y4 ``=` `2` `z4 ``=` `1` `equation_plane(x1, y1, z1, x2, y2, z2, x3,  ` `                            ``y3, z3, x4, y4, z4) `

## C#

 `// C# program to check if 4 points  ` `// in a 3-D plane are Coplanar ` `using` `System; ` ` `  `class` `GFG ` `{ ` ` `  `// Function to find equation of plane.  ` `static` `void` `equation_plane(``int` `x1, ``int` `y1, ``int` `z1, ` `                           ``int` `x2, ``int` `y2, ``int` `z2,  ` `                           ``int` `x3, ``int` `y3, ``int` `z3, ` `                           ``int` `x, ``int` `y, ``int` `z)  ` `{  ` `    ``int` `a1 = x2 - x1 ;  ` `    ``int` `b1 = y2 - y1 ;  ` `    ``int` `c1 = z2 - z1 ;  ` `    ``int` `a2 = x3 - x1 ;  ` `    ``int` `b2 = y3 - y1 ;  ` `    ``int` `c2 = z3 - z1 ;  ` `    ``int` `a = b1 * c2 - b2 * c1 ;  ` `    ``int` `b = a2 * c1 - a1 * c2 ;  ` `    ``int` `c = a1 * b2 - b1 * a2 ;  ` `    ``int` `d = (- a * x1 - b * y1 - c * z1) ;  ` `         `  `    ``// equation of plane is: a*x + b*y + c*z = 0 #  ` `         `  `    ``// checking if the 4th point satisfies  ` `    ``// the above equation  ` `    ``if``(a * x + b * y + c * z + d == 0)  ` `        ``Console.WriteLine(``"Coplanar"``);  ` `    ``else` `        ``Console.WriteLine(``"Not Coplanar"``);  ` `                     `  `}  ` `     `  `// Driver Code  ` `static` `public` `void` `Main () ` `{ ` `    ``int` `x1 = 3 ;  ` `    ``int` `y1 = 2 ;  ` `    ``int` `z1 = -5 ;  ` `    ``int` `x2 = -1 ;  ` `    ``int` `y2 = 4 ;  ` `    ``int` `z2 = -3 ;  ` `    ``int` `x3 = -3 ;  ` `    ``int` `y3 = 8 ;  ` `    ``int` `z3 = -5 ;  ` `    ``int` `x4 = -3 ;  ` `    ``int` `y4 = 2 ;  ` `    ``int` `z4 = 1 ;  ` ` `  `    ``//function calling  ` `    ``equation_plane(x1, y1, z1, x2, y2, z2,  ` `                   ``x3, y3, z3, x4, y4, z4);                          ` `}  ` `}  ` ` `  `// This code is contributed by jit_t `

## PHP

 ` `

Output:

```Coplanar
```

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