point.cpp

#include <bits/stdc++.h>

using namespace std;

#define cin_2d(vec, n, m) for(int i = 0; i < n; i++) for(int j = 0; j < m && cin >> vec[i][j]; j++);
#define cout_2d(vec, n, m) for(int i = 0; i < n; i++, cout << "\n") for(int j = 0; j < m && cout << vec[i][j] << " "; j++);
#define fixed(n) fixed << setprecision(n)
#define ceil(n, m) (((n) / (m)) + ((n) % (m) ? 1 : 0))
#define fill(vec, value) memset(vec, value, sizeof(vec));
#define mul_mod(a, b, m) (((a % m) * (b % m)) % m)
#define add_mod(a, b, m) (((a % m) + (b % m)) % m)
#define all(vec) vec.begin(), vec.end()
#define rall(vec) vec.rbegin(), vec.rend()
#define sz(x) int(x.size())
#define debug(x) cout << #x << ": " << (x) << "\n";
#define fi first
#define se second
#define ll long long
#define ull unsigned long long
#define Mod  1'000'000'007
#define OO 2'000'000'000
#define EPS 1e-9
#define PI acos(-1)
template < typename T = int > using Pair = pair < T, T >;
vector < string > RET = {"NO", "YES"};

template < typename T = int > istream& operator >> (istream &in, vector < T > &v) {
    for (auto &x : v) in >> x;
    return in;
}

template < typename T = int > ostream& operator << (ostream &out, const vector < T > &v) { 
    for (const T &x : v) out << x << ' '; 
    return out;
}

/*
 * Point — 2D geometric point with full operator overloading
 *
 * Template params:
 *   T = coordinate type (default int, use double/long double for float geometry)
 *
 * Constructor:
 *   Point<T> p(T x = 0, T y = 0);
 *   Point<T> p(const Point& other);
 *
 * Operators:
 *   +, -, *, /        → arithmetic with another point or scalar
 *   ==, !=, <, >, <=, >= → comparison (sorts by (y, x))
 *   >> , <<           → cin/cout support
 *
 * Methods:
 *   dot(p)            → T, dot product
 *   cross(p)          → T, cross product (2D)
 *   cross(a, b)       → T, cross product of vectors (a-this) and (b-this)
 *   dist()            → T, squared distance from origin
 *   dist(p)           → T, squared distance to p
 *   distance()        → double, Euclidean distance from origin
 *   distance(p)       → double, Euclidean distance to p
 *   angle()           → double, polar angle from origin (atan2)
 *   angle(p)          → double, angle between this and p
 *   unit()            → Point, unit vector (divides by dist())
 *   perp()            → Point, perpendicular vector (-y, x)
 *   rotate(a)         → Point, rotated by angle a (radians)
 *   rotate(p, a)      → Point, rotated around point p by angle a
 *   normal()          → Point, unit normal (perp().unit())
 *
 * Example:
 *   Point<ll> a(3, 4), b(0, 0);
 *   cout << a.distance(b);    // 5.0
 *   cout << a.cross(b);       // 0
 *   cout << a.perp();         // -4 3
 */
template < typename T = int > struct Point {
    T x, y;
    Point(T _x = 0, T _y = 0) : x(_x), y(_y) {}
    Point(const Point &p) : x(p.x), y(p.y) {}
    Point operator + (const Point &p) const { return Point(x + p.x, y + p.y); }
    Point operator - (const Point &p) const { return Point(x - p.x, y - p.y); }
    Point operator * (T c) const { return Point(x * c, y * c); }
    Point operator / (T c) const { return Point(x / c, y / c); }
    bool operator == (const Point &p) const { return x == p.x && y == p.y; }
    bool operator != (const Point &p) const { return x != p.x || y != p.y; }
    bool operator < (const Point &p) const { return make_pair(y, x) < make_pair(p.y, p.x); }
    bool operator > (const Point &p) const { return make_pair(y, x) > make_pair(p.y, p.x); }
    bool operator <= (const Point &p) const { return make_pair(y, x) <= make_pair(p.y, p.x); }
    bool operator >= (const Point &p) const { return make_pair(y, x) >= make_pair(p.y, p.x); }
    friend istream& operator >> (istream &in, Point &p) { return in >> p.x >> p.y; }
    friend ostream& operator << (ostream &out, const Point &p) { return out << p.x << ' ' << p.y; }
    T dot(const Point &p) const { return x * p.x + y * p.y; }
    T cross(const Point &p) const { return x * p.y - y * p.x; }
    T cross(const Point &a, const Point &b) const { return (a - *this).cross(b - *this); }
    T dist() const { return x * x + y * y; }
    T dist(const Point &p) const { return (*this - p).dist(); }
    double distance() const { return sqrt(1.0 * dist()); }
    double distance(const Point &p) const { return sqrt(1.0 * dist(p)); }
    double angle() const { return atan2(y, x); }
    double angle(const Point &p) const { return atan2(cross(p), dot(p)); }
    Point unit() const { return *this / dist(); }
    Point perp() const { return Point(-y, x); }
    Point rotate(double a) const { return Point(x * cos(a) - y * sin(a), x * sin(a) + y * cos(a)); }
    Point rotate(const Point &p, double a) const { return (*this - p).rotate(a) + p; }
    Point normal() const { return perp().unit(); }
};

void Solve(){
    
}

int main(){
    ios_base::sync_with_stdio(false), cin.tie(nullptr), cout.tie(nullptr);
    int t = 1;
    //cin >> t;
    while(t--)
        Solve();
    return 0;
}