C++ Program to Check if a Point d Lies Inside or Outside a Circle Defined by Points a, b, c in a Plane

In this articles, we implements a C++ Program to check if a point d lies inside or outside a circle defined by points a, b, c in a plane using the following equation:

s = (x-xt)^2 + (y-yt)^2 - r*r

Where equation contains the following points:

• x, y: Any point in the plane.
• xt, yt: Center of the circle.
• r: Radius of the circle.
• s: The difference between the squared distance and ( r^2 ).

If s is equal to 0, the point lies on the circle. If s is greater than 0, the point lies outside the circle, and if s is less than 0, the point lies inside the circle.

C++ program to check if a point d lies inside or outside a circle defined by Points a, b, c in a plane

In the following example, we implement a C++ program to check if a Point Lies Inside or outside a Circle:

#include <iostream>
#include <cmath>
#include <cstdlib>
#include <ctime>

using namespace std;

const int L = 0;
const int H = 20;

// Function to generate a random point within range
double randomPoint() {
   return rand() % (H - L + 1) + L;
}

int main() {
   // Seed the random number generator
   srand(static_cast < unsigned int > (time(nullptr)));

   // Define three random points on the circle
   double x1 = randomPoint(), y1 = randomPoint();
   double x2 = randomPoint(), y2 = randomPoint();
   double x3 = randomPoint(), y3 = randomPoint();

   // Calculate slopes
   double a1 = (y1 - y2) / (x1 - x2);
   double a2 = (y3 - y2) / (x3 - x2);

   // Compute center (c1, c2) using derived formulas
   double c1 = ((a1 * a2 * (y3 - y1)) + (a1 * (x2 + x3)) - (a2 * (x1 + x2))) / (2 * (a1 - a2));
   double c2 = ((((x1 + x2) / 2) - c1) / (-1 * a1)) + ((y1 + y2) / 2);

   // Compute radius
   double r = sqrt(pow(x3 - c1, 2) + pow(y3 - c2, 2));

   cout << "The points on the circle are: (" << x1 << ", " << y1 << "), (" << x2 << ", " << y2 << "), (" << x3 << ", " << y3 << ")\n";
   cout << "The center of the circle is (" << c1 << ", " << c2 << ") and radius is " << r << endl;

   double u = 10, v = 10; // Example test point
   cout << "\nTesting point (" << u << ", " << v << ")...";

   // Check position relative to the circle
   double s1 = pow(u - c1, 2) + pow(v - c2, 2) - pow(r, 2);
   if (s1 < 0)
      cout << "\nThe point lies **inside** the circle.\n";
   else if (s1 > 0)
      cout << "\nThe point lies **outside** the circle.\n";
   else
      cout << "\nThe point lies **on** the circle.\n";

   return 0;
}

The below output tells whether point d is inside or outside of circle ?

The points on the circle are: (14, 2), (4, 0), (0, 13)
The center of the circle is (7.55797, -6.21014) and radius is 20.6435

Testing point (10, 10)...
The point lies **inside** the circle.