2020ICPC·小米 網路選拔賽熱身賽K-Random Point in Triangle

#傳送門： https://ac.nowcoder.com/acm/contest/8409/K

題意

$給3個點(x1,y1),(x2,y2),(x3,y3)，在三角形內部中取一點p，設E=max(S_{PAB},S_{PAC},S_{PBC}),求E的概率期望。$

$對於任何三角形，先看看特殊的等邊三角形。然後在推廣到任意三角形（ACM需要猜！）$

$O點為三角形ABC的內心。所以對於p的位置的選取，只需要在三角形的一半，然後一半即可。即在三角形OSB中選取。$

$然後構造下圖為：$

$設該等邊三角形的邊長為2a，ON為y，NP為x，則根據相似可得：$

$PH=\frac{2}{3}a+x+\frac{\sqrt{3}}{3}y$
$PG=\frac{\sqrt{3}}{2}PH=\frac{\sqrt{3}}{3}a+\frac{\sqrt{3}}{2}x+\frac{1}{2}y$

$則E=S_{PAC}=\frac{1}{2}\ast AH\ast PG=\frac{\sqrt{3}}{3}{a}^2+\frac{\sqrt{3}}{2}ax+\frac{1}{2}ay$

$根據上圖，P點的y可在[0，\frac{\sqrt{a}}{3}a]中選取，所以x可以在[0，\sqrt{3}y]中選取。$

$對面積積分如下：$

$${\int }_0^{\frac{\sqrt{a}}{3}}dy{\int }_0^{\sqrt{3}y}(\frac{\sqrt{3}}{3}{a}^2+\frac{\sqrt{3}}{2}ax+\frac{1}{2}ay)dx=\frac{11}{36}{a}^4$$

$而P點在該區域的選取面積為\frac{\sqrt{3}}{6}{a}^2，期望為\frac{11}{18}\sqrt{3}{a}^2，而等邊三角形的面積為\sqrt{3}{a}^2$
$即期望為\frac{11}{18}{S}_{ABC}，在乘以36得22S_{ABC}。$

$把等邊三角形推廣到任意三角形，雖然我不會證明，大膽猜測總沒錯！$

#include <bits/stdc++.h>

using namespace std;

typedef long long ll;
typedef long double ld;
typedef pair<int, int> pdd;

#define INF 0x3f3f3f3f
#define lowbit(x) x & (-x)
#define mem(a, b) memset(a , b , sizeof(a))
#define FOR(i, x, n) for(int i = x;i <= n; i++)

// const ll mod = 998244353;
 const ll mod = 1e9 + 7;
// const double eps = 1e-6;
// const double PI = acos(-1);
// const double R = 0.57721566490153286060651209;

void solve() {
    ll x1, x2, x3, y1, y2, y3;
    while(cin >> x1 >> y1 >> x2 >> y2 >> x3 >> y3) {
        cout <<  abs(x1 * y2 - x2 * y1 + x3 * y1 - x1 * y3 + x2 * y3 - x3 * y2) * 11 << endl;
    }
}

signed main() {
    ios_base::sync_with_stdio(false);
    //cin.tie(nullptr);
    //cout.tie(nullptr);
#ifdef FZT_ACM_LOCAL
    freopen("in.txt", "r", stdin);
    freopen("out.txt", "w", stdout);
    signed test_index_for_debug = 1;
    char acm_local_for_debug = 0;
    do {
        if (acm_local_for_debug == '$') exit(0);
        if (test_index_for_debug > 20)
            throw runtime_error("Check the stdin!!!");
        auto start_clock_for_debug = clock();
        solve();
        auto end_clock_for_debug = clock();
        cout << "Test " << test_index_for_debug << " successful" << endl;
        cerr << "Test " << test_index_for_debug++ << " Run Time: "
             << double(end_clock_for_debug - start_clock_for_debug) / CLOCKS_PER_SEC << "s" << endl;
        cout << "--------------------------------------------------" << endl;
    } while (cin >> acm_local_for_debug && cin.putback(acm_local_for_debug));
#else
    solve();
#endif
    return 0;
}