# Number of possible Triangles in a Cartesian coordinate system

Given n points in a Cartesian coordinate system. Count the number of triangles formed.

Examples:

Input  : point[] = {(0, 0), (1, 1), (2, 0), (2, 2)
Output : 3
Three triangles can be formed from above points.


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

A simple solution is to check if the determinant of the three points selected is non-zero or not. The following determinant gives the area of a Triangle (Also known as Cramer’s rule).

Area of the triangle with corners at (x1, y1), (x2, y2) and (x3, y3) is given by: We can solve this by taking all possible combination of 3 points and finding the determinant.

## C++

 // C++ program to count number of triangles that can  // be formed with given points in 2D  #include  using namespace std;     // A point in 2D  struct Point  {     int x, y;  };     // Returns determinant value of three points in 2D  int det(int x1, int y1, int x2, int y2, int x3, int y3)  {     return x1*(y2 - y3) - y1*(x2 - x3) + 1*(x2*y3 - y2*x3);  }     // Returns count of possible triangles with given array  // of points in 2D.  int countPoints(Point arr[], int n)  {      int result = 0;  // Initialize result         // Consider all triplets of points given in inputs      // Increment the result when determinant of a triplet      // is not 0.      for (int i=0; i

## Java

 // Java program to count number   // of triangles that can be formed   // with given points in 2D     class GFG{  // Returns determinant value   // of three points in 2D  static int det(int x1, int y1, int x2, int y2, int x3, int y3)  {      return (x1 * (y2 - y3) - y1 *          (x2 - x3) + 1 * (x2 *              y3 - y2 * x3));  }     // Returns count of possible   // triangles with given array  // of points in 2D.  static int countPoints(int [][]Point, int n)  {      int result = 0; // Initialize result         // Consider all triplets of       // points given in inputs      // Increment the result when       // determinant of a triplet is not 0.      for(int i = 0; i < n; i++)          for(int j = i + 1; j < n; j++)              for(int k = j + 1; k < n; k++)                  if(det(Point[i], Point[i],                       Point[j], Point[j],                       Point[k], Point[k])>=0)                      result = result + 1;         return result;  }     // Driver code  public static void main(String[] args)  {  int Point[][] = {{0, 0},{1, 1},{2, 0},{2, 2}};  int n = Point.length;  System.out.println(countPoints(Point, n));  }  }  // This code is contributed by  // mits 

## Python

 # Python program to count number   # of triangles that can be formed   # with given points in 2D     # Returns determinant value   # of three points in 2D  def det(x1, y1, x2, y2, x3, y3):      return (x1 * (y2 - y3) - y1 *              (x2 - x3) + 1 * (x2 *              y3 - y2 * x3))     # Returns count of possible   # triangles with given array  # of points in 2D.  def countPoints(Point, n):         result = 0 # Initialize result         # Consider all triplets of       # points given in inputs      # Increment the result when       # determinant of a triplet is not 0.      for i in range(n):          for j in range(i + 1, n):              for k in range(j + 1, n):                  if(det(Point[i], Point[i],                          Point[j], Point[j],                          Point[k], Point[k])):                      result = result + 1        return result     # Driver code  Point = [[0, 0], [1, 1],            [2, 0], [2, 2]]  n = len(Point)  print(countPoints(Point, n))     # This code is contributed by  # Sanjit_Prasad 

## C#

 // C# program to count number   // of triangles that can be formed   // with given points in 2D  using System;     class GFG{  // Returns determinant value   // of three points in 2D  static int det(int x1, int y1, int x2, int y2, int x3, int y3)  {      return (x1 * (y2 - y3) - y1 *          (x2 - x3) + 1 * (x2 *              y3 - y2 * x3));  }     // Returns count of possible   // triangles with given array  // of points in 2D.  static int countPoints(int[,] Point, int n)  {      int result = 0; // Initialize result         // Consider all triplets of       // points given in inputs      // Increment the result when       // determinant of a triplet is not 0.      for(int i = 0; i < n; i++)          for(int j = i + 1; j < n; j++)              for(int k = j + 1; k < n; k++)                  if(det(Point[i,0], Point[i,1], Point[j,0], Point[j,1],Point[k,0], Point[k,1])>=0)                      result = result + 1;         return result;  }     // Driver code  public static void Main()  {  int[,] Point = new int[,] { { 0, 0 }, { 1, 1 }, { 2, 0 }, { 2, 2 } };  int n = Point.Length/Point.Rank;  Console.WriteLine(countPoints(Point, n));  }  }  // This code is contributed by mits 

## PHP

  

Output:

3
`

Time Complexity: .

Optimization :
We can optimize the above solution to work in O(n2) using the fact that three points cannot form a triangle if they are collinear. We can use hashing to store slopes of all pairs and find all triangles in O(n2) time.

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