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
1 #include <bits/stdc++.h> 2 using namespace std; 3 4 #define INIT() ios::sync_with_stdio(false);cin.tie(0);cout.tie(0); 5 #define Rep(i,n) for (int i = 0; i < (n); ++i) 6 #define For(i,s,t) for (int i = (s); i <= (t); ++i) 7 #define rFor(i,t,s) for (int i = (t); i >= (s); --i) 8 #define ForLL(i, s, t) for (LL i = LL(s); i <= LL(t); ++i) 9 #define rForLL(i, t, s) for (LL i = LL(t); i >= LL(s); --i) 10 #define foreach(i,c) for (__typeof(c.begin()) i = c.begin(); i != c.end(); ++i) 11 #define rforeach(i,c) for (__typeof(c.rbegin()) i = c.rbegin(); i != c.rend(); ++i) 12 13 #define pr(x) cout << #x << " = " << x << " " 14 #define prln(x) cout << #x << " = " << x << endl 15 16 #define LOWBIT(x) ((x)&(-x)) 17 18 #define ALL(x) x.begin(),x.end() 19 #define INS(x) inserter(x,x.begin()) 20 #define UNIQUE(x) x.erase(unique(x.begin(), x.end()), x.end()) 21 #define REMOVE(x, c) x.erase(remove(x.begin(), x.end(), c), x.end()); // ?? x ?????? c 22 #define TOLOWER(x) transform(x.begin(), x.end(), x.begin(),::tolower); 23 #define TOUPPER(x) transform(x.begin(), x.end(), x.begin(),::toupper); 24 25 #define ms0(a) memset(a,0,sizeof(a)) 26 #define msI(a) memset(a,inf,sizeof(a)) 27 #define msM(a) memset(a,-1,sizeof(a)) 28 29 #define MP make_pair 30 #define PB push_back 31 #define ft first 32 #define sd second 33 34 template<typename T1, typename T2> 35 istream &operator>>(istream &in, pair<T1, T2> &p) { 36 in >> p.first >> p.second; 37 return in; 38 } 39 40 template<typename T> 41 istream &operator>>(istream &in, vector<T> &v) { 42 for (auto &x: v) 43 in >> x; 44 return in; 45 } 46 47 template<typename T> 48 ostream &operator<<(ostream &out, vector<T> &v) { 49 Rep(i, v.size()) out << v[i] << " "[i == v.size()]; 50 return out; 51 } 52 53 template<typename T1, typename T2> 54 ostream &operator<<(ostream &out, const std::pair<T1, T2> &p) { 55 out << "[" << p.first << ", " << p.second << "]" << " "; 56 return out; 57 } 58 59 inline int gc(){ 60 static const int BUF = 1e7; 61 static char buf[BUF], *bg = buf + BUF, *ed = bg; 62 63 if(bg == ed) fread(bg = buf, 1, BUF, stdin); 64 return *bg++; 65 } 66 67 inline int ri(){ 68 int x = 0, f = 1, c = gc(); 69 for(; c<48||c>57; f = c=='-'?-1:f, c=gc()); 70 for(; c>47&&c<58; x = x*10 + c - 48, c=gc()); 71 return x*f; 72 } 73 74 template<class T> 75 inline string toString(T x) { 76 ostringstream sout; 77 sout << x; 78 return sout.str(); 79 } 80 81 inline int toInt(string s) { 82 int v; 83 istringstream sin(s); 84 sin >> v; 85 return v; 86 } 87 88 //min <= aim <= max 89 template<typename T> 90 inline bool BETWEEN(const T aim, const T min, const T max) { 91 return min <= aim && aim <= max; 92 } 93 94 typedef long long LL; 95 typedef unsigned long long uLL; 96 typedef pair< double, double > PDD; 97 typedef pair< int, int > PII; 98 typedef pair< int, PII > PIPII; 99 typedef pair< string, int > PSI; 100 typedef pair< int, PSI > PIPSI; 101 typedef set< int > SI; 102 typedef set< PII > SPII; 103 typedef vector< int > VI; 104 typedef vector< double > VD; 105 typedef vector< VI > VVI; 106 typedef vector< SI > VSI; 107 typedef vector< PII > VPII; 108 typedef map< int, int > MII; 109 typedef map< int, string > MIS; 110 typedef map< int, PII > MIPII; 111 typedef map< PII, int > MPIII; 112 typedef map< string, int > MSI; 113 typedef map< string, string > MSS; 114 typedef map< PII, string > MPIIS; 115 typedef map< PII, PII > MPIIPII; 116 typedef multimap< int, int > MMII; 117 typedef multimap< string, int > MMSI; 118 //typedef unordered_map< int, int > uMII; 119 typedef pair< LL, LL > PLL; 120 typedef vector< LL > VL; 121 typedef vector< VL > VVL; 122 typedef priority_queue< int > PQIMax; 123 typedef priority_queue< int, VI, greater< int > > PQIMin; 124 const double EPS = 1e-8; 125 const LL inf = 0x7fffffff; 126 const LL infLL = 0x7fffffffffffffffLL; 127 const LL mod = 1e9 + 7; 128 const int maxN = 2e5 + 7; 129 const LL ONE = 1; 130 const LL evenBits = 0xaaaaaaaaaaaaaaaa; 131 const LL oddBits = 0x5555555555555555; 132 133 template<typename T> 134 struct Point{ 135 T X,Y; 136 Point< T >(T x = 0,T y = 0) : X(x), Y(y) {} 137 138 inline Point<T> operator* (const T& k) { return Point<T>(k*X, k*Y); } 139 inline Point<T> operator/ (const T& k) { return Point<T>(X/k, Y/k); } 140 inline Point<T> operator+ (const Point<T>& x) { return Point<T>(x.X+X,x.Y+Y); } 141 inline Point<T> operator- (const Point<T>& x) { return Point<T>(x.X-X,x.Y-Y); } 142 143 T operator^(const Point<T> &x) const { return X*x.Y - Y*x.X; } 144 T operator*(const Point<T> &x) const { return X*x.X + Y*x.Y; } 145 }; 146 147 template<typename T> 148 istream &operator>> (istream &in, Point<T> &x) { 149 in >> x.X >> x.Y; 150 return in; 151 } 152 153 Point< LL > p[3]; 154 155 int main(){ 156 //freopen("MyOutput.txt","w",stdout); 157 //freopen("input.txt","r",stdin); 158 //INIT(); 159 while(cin >> p[0] >> p[1] >> p[2]) cout << 11 * abs((p[0] - p[1]) ^ (p[0] - p[2])) << endl; 160 return 0; 161 }