It's always a hassle to define our 2D Geometry library during a contest. Is there a way to make our computational geometry lives easier in any way? Fortunately for us, there is, at least in C++, using complex numbers.

Complex numbers are of the form a + bi, where a is the real part and b imaginary. Thus, we can let a be the x-coordinate and b be the y-coordinate. Whelp, complex numbers can be represented as 2D vectors! Therefore, we can use complex numbers to define a point instead of defining the class ourselves. You can look at std::complex reference here.

Defining our point class

We can define our point class by typing typedef complex<double> point; at the start of our program. To access our x- and y-coordinates, we can macro the real() and imag() functions by using #define. Of course, don't forget to #include <complex> before anything.

#include <iostream>
#include <complex>
using namespace std;

// define x, y as real(), imag()
typedef complex<double> point;
#define x real()
#define y imag()

// sample program
int main() {
	point a = 2;
	point b(3, 7);
	cout << a << ' ' << b << endl; // (2, 0) (3, 7)
	cout << a + b << endl;         // (5, 7)
}

Oh goodie! We can use std:cout for debugging! We can also add points as vectors without having to define operator+. Nifty. And apparently, we can overall add points, subtract points, do scalar multiplication without defining any operator. Very nifty indeed.

Example

point a(3, 2), b(2, -7);

// vector addition and subtraction
cout << a + b << endl;   // (5,-5)
cout << a - b << endl;   // (1,9)

// scalar multiplication
cout << 3.0 * a << endl; // (9,6)
cout << a / 5.0 << endl; // (0.6,0.4)

Functions using std::complex

What else can we do with complex numbers? Well, there's a lot that is really easy to code.

1. Vector addition: a + b

2. Scalar multiplication: r * a

3. Dot product: (conj(a) * b).x

4. Cross product: (conj(a) * b).y
   
   notice: conj(a) * b = (ax*bx + ay*by) + i (ax*by — ay*bx)

5. Squared distance: norm(a - b)

6. Euclidean distance: abs(a - b)

7. Angle of elevation: arg(b - a)

8. Slope of line (a, b): tan(arg(b - a))

9. Polar to cartesian: polar(r, theta)

10. Cartesian to polar: point(abs(p), arg(p))

11. Rotation about the origin: a * polar(1.0, theta)

12. Rotation about pivot p: (a-p) * polar(1.0, theta) + p
    
    UPD: added more useful functions

13. Angle ABC: abs(remainder(arg(a-b) - arg(c-b), 2.0 * M_PI))
    
    remainder normalizes the angle to be between [-PI, PI]. Thus, we can get the positive non-reflex angle by taking its abs value.

14. Project p onto vector v: v * dot(p, v) / norm(v);

15. Project p onto line (a, b): a + (b - a) * dot(p - a, b - a) / norm(b - a)

16. Reflect p across line (a, b): a + conj((p - a) / (b - a)) * (b - a)

17. Intersection of line (a, b) and (p, q):

point intersection(point a, point b, point p, point q) {
  double c1 = cross(p - a, b - a), c2 = cross(q - a, b - a);
  return (c1 * q - c2 * p) / (c1 - c2); // undefined if parallel
}

Drawbacks

Using std::complex is very advantageous, but it has one disadvantage: you can't use std::cin or scanf. Also, if we macro x and y, we can't use them as variables. But that's rather minor, don't you think?

EDIT: Credits to Zlobober for pointing out that std::complex has issues with integral data types. The library will work for simple arithmetic like vector addition and such, but not for polar or abs. It will compile but there will be some errors in correctness! The tip then is to rely on the library only if you're using floating point data all throughout.