# How to check if given four points form a square

Given coordinates of four points in a plane, find if the four points form a square or not.

To check for square, we need to check for following.

a) All fours sides formed by points are the same.

b) The angle between any two sides is 90 degree. (This condition is required as Quadrilateral also has same sides.

c) Check both the diagonals have the same distance

The idea is to pick any point and calculate its distance from the rest of the points. Let the picked point be ‘p’. To form a square, the distance of two points must be the same from ‘p’, let this distance be d. The distance from one point must be different from that d and must be equal to √2 times d. Let this point with different distance be ‘q’.

The above condition is not good enough as the point with different distance can be on the other side. We also need to check that q is at the same distance from 2 other points and this distance is the same as d.

Below are the implementations of the above idea.

## C++

// A C++ program to check if four given points form a square or not. #include <iostream> using namespace std; // Structure of a point in 2D space struct Point { int x, y; }; // A utility function to find square of distance // from point 'p' to point 'q' int distSq(Point p, Point q) { return (p.x - q.x) * (p.x - q.x) + (p.y - q.y) * (p.y - q.y); } // This function returns true if (p1, p2, p3, p4) form a // square, otherwise false bool isSquare(Point p1, Point p2, Point p3, Point p4) { int d2 = distSq(p1, p2); // from p1 to p2 int d3 = distSq(p1, p3); // from p1 to p3 int d4 = distSq(p1, p4); // from p1 to p4 if (d2 == 0 || d3 == 0 || d4 == 0) return false; // If lengths if (p1, p2) and (p1, p3) are same, then // following conditions must met to form a square. // 1) Square of length of (p1, p4) is same as twice // the square of (p1, p2) // 2) Square of length of (p2, p3) is same // as twice the square of (p2, p4) if (d2 == d3 && 2 * d2 == d4 && 2 * distSq(p2, p4) == distSq(p2, p3)) { return true; } // The below two cases are similar to above case if (d3 == d4 && 2 * d3 == d2 && 2 * distSq(p3, p2) == distSq(p3, p4)) { return true; } if (d2 == d4 && 2 * d2 == d3 && 2 * distSq(p2, p3) == distSq(p2, p4)) { return true; } return false; } // Driver program to test above function int main() { Point p1 = { 20, 10 }, p2 = { 10, 20 }, p3 = { 20, 20 }, p4 = { 10, 10 }; isSquare(p1, p2, p3, p4) ? cout << "Yes" : cout << "No"; return 0; }

## Java

// A Java program to check if four given points form a square or not. class GFG { // Structure of a point in 2D space static class Point { int x, y; public Point(int x, int y) { this.x = x; this.y = y; } }; // A utility function to find square of distance // from point 'p' to point 'q' static int distSq(Point p, Point q) { return (p.x - q.x) * (p.x - q.x) + (p.y - q.y) * (p.y - q.y); } // This function returns true if (p1, p2, p3, p4) form a // square, otherwise false static boolean isSquare(Point p1, Point p2, Point p3, Point p4) { int d2 = distSq(p1, p2); // from p1 to p2 int d3 = distSq(p1, p3); // from p1 to p3 int d4 = distSq(p1, p4); // from p1 to p4 if (d2 == 0 || d3 == 0 || d4 == 0) return false; // If lengths if (p1, p2) and (p1, p3) are same, then // following conditions must met to form a square. // 1) Square of length of (p1, p4) is same as twice // the square of (p1, p2) // 2) Square of length of (p2, p3) is same // as twice the square of (p2, p4) if (d2 == d3 && 2 * d2 == d4 && 2 * distSq(p2, p4) == distSq(p2, p3)) { return true; } // The below two cases are similar to above case if (d3 == d4 && 2 * d3 == d2 && 2 * distSq(p3, p2) == distSq(p3, p4)) { return true; } if (d2 == d4 && 2 * d2 == d3 && 2 * distSq(p2, p3) == distSq(p2, p4)) { return true; } return false; } // Driver code public static void main(String[] args) { Point p1 = new Point(20, 10), p2 = new Point( 10, 20 ), p3 = new Point(20, 20 ), p4 = new Point( 10, 10 ); System.out.println(isSquare(p1, p2, p3, p4)==true ? "Yes" : "No"); } } // This code is contributed by PrinciRaj1992

## Python3

# A Python3 program to check if # four given points form a square or not. class Point: # Structure of a point in 2D space def __init__(self, x, y): self.x = x self.y = y # A utility function to find square of # distance from point 'p' to point 'q' def distSq(p, q): return (p.x - q.x) * (p.x - q.x) +\ (p.y - q.y) * (p.y - q.y) # This function returns true if (p1, p2, p3, p4) # form a square, otherwise false def isSquare(p1, p2, p3, p4): d2 = distSq(p1, p2) # from p1 to p2 d3 = distSq(p1, p3) # from p1 to p3 d4 = distSq(p1, p4) # from p1 to p4 if d2 == 0 or d3 == 0 or d4 == 0: return False # If lengths if (p1, p2) and (p1, p3) are same, then # following conditions must be met to form a square. # 1) Square of length of (p1, p4) is same as twice # the square of (p1, p2) # 2) Square of length of (p2, p3) is same # as twice the square of (p2, p4) if d2 == d3 and 2 * d2 == d4 and \ 2 * distSq(p2, p4) == distSq(p2, p3): return True # The below two cases are similar to above case if d3 == d4 and 2 * d3 == d2 and \ 2 * distSq(p3, p2) == distSq(p3, p4): return True if d2 == d4 and 2 * d2 == d3 and \ 2 * distSq(p2, p3) == distSq(p2, p4): return True return False # Driver Code if __name__=="__main__": p1 = Point(20, 10) p2 = Point(10, 20) p3 = Point(20, 20) p4 = Point(10, 10) if isSquare(p1, p2, p3, p4): print('Yes') else: print('No') # This code is contributed by Mayank Chaudhary # aka chaudhary_19

## C#

// A C# program to check if four given points form a square or not. using System; class GFG { // Structure of a point in 2D space class Point { public int x, y; public Point(int x, int y) { this.x = x; this.y = y; } }; // A utility function to find square of distance // from point 'p' to point 'q' static int distSq(Point p, Point q) { return (p.x - q.x) * (p.x - q.x) + (p.y - q.y) * (p.y - q.y); } // This function returns true if (p1, p2, p3, p4) form a // square, otherwise false static bool isSquare(Point p1, Point p2, Point p3, Point p4) { int d2 = distSq(p1, p2); // from p1 to p2 int d3 = distSq(p1, p3); // from p1 to p3 int d4 = distSq(p1, p4); // from p1 to p4 if (d2 == 0 || d3 == 0 || d4 == 0) return false; // If lengths if (p1, p2) and (p1, p3) are same, then // following conditions must met to form a square. // 1) Square of length of (p1, p4) is same as twice // the square of (p1, p2) // 2) Square of length of (p2, p3) is same // as twice the square of (p2, p4) if (d2 == d3 && 2 * d2 == d4 && 2 * distSq(p2, p4) == distSq(p2, p3)) { return true; } // The below two cases are similar to above case if (d3 == d4 && 2 * d3 == d2 && 2 * distSq(p3, p2) == distSq(p3, p4)) { return true; } if (d2 == d4 && 2 * d2 == d3 && 2 * distSq(p2, p3) == distSq(p2, p4)) { return true; } return false; } // Driver code public static void Main(String[] args) { Point p1 = new Point(20, 10), p2 = new Point(10, 20), p3 = new Point(20, 20), p4 = new Point(10, 10); Console.WriteLine(isSquare(p1, p2, p3, p4) == true ? "Yes" : "No"); } } // This code is contributed by 29AjayKumar

## Javascript

<script> // JavaScript program to check if four given points form a square or not. // A utility function to find square of distance // from point 'p' to point 'q' function distSq( p, q){ return (p.x - q.x) * (p.x - q.x) + (p.y - q.y) * (p.y - q.y); } // This function returns true if (p1, p2, p3, p4) form a // square, otherwise false function isSquare(p1, p2, p3, p4){ let d2 = distSq(p1, p2); // from p1 to p2 let d3 = distSq(p1, p3); // from p1 to p3 let d4 = distSq(p1, p4); // from p1 to p4 if (d2 == 0 || d3 == 0 || d4 == 0) return false; // If lengths if (p1, p2) and (p1, p3) are same, then // following conditions must met to form a square. // 1) Square of length of (p1, p4) is same as twice // the square of (p1, p2) // 2) Square of length of (p2, p3) is same // as twice the square of (p2, p4) if (d2 == d3 && 2 * d2 == d4 && 2 * distSq(p2, p4) == distSq(p2, p3)) { return true; } // The below two cases are similar to above case if (d3 == d4 && 2 * d3 == d2 && 2 * distSq(p3, p2) == distSq(p3, p4)) { return true; } if (d2 == d4 && 2 * d2 == d3 && 2 * distSq(p2, p3) == distSq(p2, p4)) { return true; } return false; } // Driver program to test above function let p1 = { x:20, y:10 } let p2 = { x:10, y:20 } let p3 = { x:20, y:20 } let p4 = { x:10, y:10 } isSquare(p1, p2, p3, p4) ? document.write("Yes") : document.write("No"); // This code is contributed by rohitsingh07052. </script>

**Output:**

Yes

**Extended Problem:**

Check if four segments form a rectangle

This article is contributed by **Anuj**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above