Point.h

#include <algorithm>
#include <cmath>
using namespace std;



#define sz(x) (int)(x).size()
#define all(x) x.begin(),x.end()


template<class T>
struct Point
{
    typedef Point P;
    /* data */
    T x,y;
    explicit Point(T x=0, T y=0):x(x), y(y){}

    bool operator<(P p)const{return tie(x,y) < tie(p.x,p.y);}
    bool operator==(P p)const{return tie(x,y) == tie(p.x,p.y);}
    P operator+(P p) const{return P(x+p.x, y+p.y);}
    P operator-(P p) const{return P(x-p.x, y-p.y);}
    P operator*(T d) const{return P(x*d, y*d);}
    P operator/(T d) const{return P(x/d, y/d);}

    T dot(P p)const{return x*p.x + y*p.y;}
    T cross(P p)const{return x*p.y - y*p.x;}
    T cross(P a, P b)const{return (a - *this).cross(b-*this);}

    T dist2() const {return x*x + y*y;}

    double dist()const{return sqrt((double)dist2());}

    double angle() const {return atan2(y,x);}

    P unit() const {return *this/dist();}
    P perp() const {return P(-y,x);}

    P normal() const {return perp().unit();}
    P rotate(double a) const {
        return P(x*cos(a) - y*sin(a), x*sin(a) + y*cos(a));
    } 

};