How to check if four points form a square

Understand the problem

Assume that we get four coordinates of points in the 2d plane. To check for the squareness of a given point, we must verify the following.

Each of the four sides created by points is identical.
Any two sides form an angle of 90 degrees.
Verify that both diagonals have the same length.

Now to know the distance between 2 points, we have the following formula:

$AB = \sqrt{(x_2−x_1)^2 + (y_2−y_1)^2}$

Solution

The objective is to find the distance between any point and the remaining points. Let the selected point be "p". To make a square, the distance between two points from "p" must be the same. Denote this distance as d. The distance between two points must be distinct from d and equal to $\sqrt{2}$ times d. Let q represent this point with a different distance.

Additionally, we must ensure that q is at the same distance from two other points and that this distance equals d.

Implementation

The implementation of the solution explained above in C++ is given below:

#include<iostream>
using namespace std;
struct Point {
   int x, y;
};
int squareDist(Point p, Point q) { //square distance formula
   return (p.x - q.x)*(p.x - q.x) + (p.y - q.y)*(p.y - q.y);
}
bool isSquare(Point p1, Point p2, Point p3, Point p4) {    //check four points are forming square or not
   int dist2 = squareDist(p1, p2);     // distance from p1 to p2
   int dist3 = squareDist(p1, p3);     // distance from p1 to p3
   int dist4 = squareDist(p1, p4);     // distance from p1 to p4
   //when length of p1-p2 and p1-p3 are same, and square of (p1-p4) = 2*(p1-p2)
   if (dist2 == dist3 && 2*dist2 == dist4) {
      int dist = squareDist(p2, p4);
      return (dist == squareDist(p3, p4) && dist == dist2);
   }
   //same condition for all other combinations
   if (dist3 == dist4 && 2*dist3 == dist2) {
      int dist = squareDist(p2, p3);
      return (dist == squareDist(p2, p4) && dist == dist3);
   }
   if (dist2 == dist4 && 2*dist2 == dist3) {
      int dist = squareDist(p2, p3);
      return (dist == squareDist(p3, p4) && dist == dist2);
  }
  return false;
}
int main() {
   Point p1 = {20, 10}, p2 = {10, 20}, p3 = {20, 20}, p4 = {10, 10};
   if(isSquare(p1, p2, p3, p4))
      cout << "Points are forming a square.";
   else
      cout << "Points are not forming a square";
}

Code explanation

Lines 4–9: We create the struct Point for the 2d coordinates in the x,y plane. We define the squareDist() formula to calculate the distance between two points.
Lines 12–15: We define isSquare() function to check whether given points make a square or not. We initialize three variables to calculate the distance of p1 from p2,p3 and p4 .
Lines 17–21: For dist2, we check if it is equal to the dist3 and two times of dist4. We return true if the dist is equal to the dist3 and the squareDist() of other points.
Lines 23–33: We define the same condition for the remaining points.

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