2019 牛客多校第一场 F Random Point in Triangle

题目链接：https://ac.nowcoder.com/acm/contest/881/F

题目大意

　　给定二维平面上 3 个整数表示的点 A，B，C，在三角形 ABC 内随机选一点 P，求期望$E = max(S_{PAB}, S_{PAC}, S_{PBC})$。输出 36 * E。

分析

　　先说结论，答案是$22S_{ABC}$，证明如下：

　　不妨设 A 为 (0, 0)，B 为 (1, 0), C 为 (a, b)，这是因为对于任意一个三角形，总可以把 A 点移动到原点，然后旋转使 AB 与 x 轴重合，然后缩放使 AB = 1。

　　设 P 为 (x, y)。

　　设重心为 G，各点对应边中点分别为 A', B', C'。

　　由于$S_{PAB}, S_{PAC}, S_{PBC}$地位是相等的，因此下面只讨论$E = S_{PAB}$的情况，也就是 P 落在四边形 CB'GA' 内部的情况。

　　简图如下：

 

　　以下列出一些点的坐标和一些直线的函数方程：

• G：$(\frac{a + 1}{3}, \frac{b}{3})$
• B'：$(\frac{a}{2}, \frac{b}{2})$
• A'：$(\frac{a + 1}{2}, \frac{b}{2})$
• B'A'：$x = \frac{b}{2}$
• CA'：$y = \frac{b}{a - 1}(x - 1)$
• CB'：$y = \frac{b}{a}x$
• GB'：$y = \frac{b}{a - 2}(x - 1)$
• GA'：$y = \frac{b}{a + 1}x$
• $S_{PAB}$：$\frac{y}{2}$
• $S_{CB'GA'}$：$\frac{b}{6}$
• $S_{CB'A'}$：$\frac{b}{8}$
• $S_{GB'A'}$：$\frac{b}{24}$

　　于是$E = \frac{1}{S_{CB'GA'}} ((\int_{G_y}^{A'B'} dy \int_{GB'}^{GA'} S_{PAB} dx) + (\int_{A'B'}^{C_y} dy \int_{CB'}^{CA'} S_{PAB} dx))$

　　积出来得$E = \frac{11b}{36} = \frac{11}{18}S_{ABC}$

　　所以$36E = 22S_{ABC}$

代码如下

 #include <bits/stdc++.h>
 using namespace std;

 #define INIT() ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
 #define Rep(i,n) for (int i = 0; i < (n); ++i)
 #define For(i,s,t) for (int i = (s); i <= (t); ++i)
 #define rFor(i,t,s) for (int i = (t); i >= (s); --i)
 #define ForLL(i, s, t) for (LL i = LL(s); i <= LL(t); ++i)
 #define rForLL(i, t, s) for (LL i = LL(t); i >= LL(s); --i)
 #define foreach(i,c) for (__typeof(c.begin()) i = c.begin(); i != c.end(); ++i)
 #define rforeach(i,c) for (__typeof(c.rbegin()) i = c.rbegin(); i != c.rend(); ++i)

 #define pr(x) cout << #x << " = " << x << "  "
 #define prln(x) cout << #x << " = " << x << endl

 #define LOWBIT(x) ((x)&(-x))

 #define ALL(x) x.begin(),x.end()
 #define INS(x) inserter(x,x.begin())
 #define UNIQUE(x) x.erase(unique(x.begin(), x.end()), x.end())
 #define REMOVE(x, c) x.erase(remove(x.begin(), x.end(), c), x.end()); // ?? x ?????? c
 #define TOLOWER(x) transform(x.begin(), x.end(), x.begin(),::tolower);
 #define TOUPPER(x) transform(x.begin(), x.end(), x.begin(),::toupper);

 #define ms0(a) memset(a,0,sizeof(a))
 #define msI(a) memset(a,inf,sizeof(a))
 #define msM(a) memset(a,-1,sizeof(a))

 #define MP make_pair
 #define PB push_back
 #define ft first
 #define sd second

 template<typename T1, typename T2>
 istream &operator>>(istream &in, pair<T1, T2> &p) {
     in >> p.first >> p.second;
     return in;
 }

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

 template<typename T>
 ostream &operator<<(ostream &out, vector<T> &v) {
     Rep(i, v.size()) out << v[i] << " \n"[i == v.size()];
     return out;
 }

 template<typename T1, typename T2>
 ostream &operator<<(ostream &out, const std::pair<T1, T2> &p) {
     out << "[" << p.first << ", " << p.second << "]" << "\n";
     return out;
 }

 inline int gc(){
     static const int BUF = 1e7;
     static char buf[BUF], *bg = buf + BUF, *ed = bg;

     if(bg == ed) fread(bg = buf, , BUF, stdin);
     return *bg++;
 } 

 inline int ri(){
     int x = , f = , c = gc();
     for(; c<||c>; f = c=='-'?-:f, c=gc());
     for(; c>&&c<; x = x* + c - , c=gc());
     return x*f;
 }

 template<class T>
 inline string toString(T x) {
     ostringstream sout;
     sout << x;
     return sout.str();
 }

 inline int toInt(string s) {
     int v;
     istringstream sin(s);
     sin >> v;
     return v;
 }

 //min <= aim <= max
 template<typename T>
 inline bool BETWEEN(const T aim, const T min, const T max) {
     return min <= aim && aim <= max;
 }

 typedef long long LL;
 typedef unsigned long long uLL;
 typedef pair< double, double > PDD;
 typedef pair< int, int > PII;
 typedef pair< int, PII > PIPII;
 typedef pair< string, int > PSI;
 typedef pair< int, PSI > PIPSI;
 typedef set< int > SI;
 typedef set< PII > SPII;
 typedef vector< int > VI;
 typedef vector< double > VD;
 typedef vector< VI > VVI;
 typedef vector< SI > VSI;
 typedef vector< PII > VPII;
 typedef map< int, int > MII;
 typedef map< int, string > MIS;
 typedef map< int, PII > MIPII;
 typedef map< PII, int > MPIII;
 typedef map< string, int > MSI;
 typedef map< string, string > MSS;
 typedef map< PII, string > MPIIS;
 typedef map< PII, PII > MPIIPII;
 typedef multimap< int, int > MMII;
 typedef multimap< string, int > MMSI;
 //typedef unordered_map< int, int > uMII;
 typedef pair< LL, LL > PLL;
 typedef vector< LL > VL;
 typedef vector< VL > VVL;
 typedef priority_queue< int > PQIMax;
 typedef priority_queue< int, VI, greater< int > > PQIMin;
 const double EPS = 1e-;
 const LL inf = 0x7fffffff;
 const LL infLL = 0x7fffffffffffffffLL;
 const LL mod = 1e9 + ;
 const int maxN = 2e5 + ;
 const LL ONE = ;
 const LL evenBits = 0xaaaaaaaaaaaaaaaa;
 const LL oddBits = 0x5555555555555555;

 template<typename T>
 struct Point{
     T X,Y;
     Point< T >(T x = ,T y = ) : X(x), Y(y) {}

     inline Point<T> operator* (const T& k) { return Point<T>(k*X, k*Y); }
     inline Point<T> operator/ (const T& k) { return Point<T>(X/k, Y/k); }
     inline Point<T> operator+ (const Point<T>& x) { return Point<T>(x.X+X,x.Y+Y); }
     inline Point<T> operator- (const Point<T>& x) { return Point<T>(x.X-X,x.Y-Y); }

     T operator^(const Point<T> &x) const { return X*x.Y - Y*x.X; }
     T operator*(const Point<T> &x) const { return X*x.X + Y*x.Y; }
 };

 template<typename T>
 istream &operator>> (istream &in, Point<T> &x) {
     in >> x.X >> x.Y;
     return in;
 }

 Point< LL > p[];

 int main(){
     //freopen("MyOutput.txt","w",stdout);
     //freopen("input.txt","r",stdin);
     //INIT();
     while(cin >> p[] >> p[] >> p[]) cout <<  * abs((p[] - p[]) ^ (p[] - p[])) << endl;
     return ;
 }