#include using namespace std; #define pb push_back #define rep(i,n) for(long long i = 0; i < (long long)(n); i++) typedef double number; const number eps = 1e-8; using arr = vector; using matrix = vector; ostream &operator<<(ostream &o, const arr &v) { for (int i = 0; i < v.size(); i++) { cout << v[i] << " "; } cout << endl; return o; } ostream &operator<<(ostream &o, const matrix &v) { for (int i = 0; i < v.size(); i++) { cout << v[i]; } return o; } // O( n^2 ) matrix zero(int n) { matrix A(n, arr(n, 0)); return A; } // O( n^2 ) matrix identity(int n) { matrix A(n, arr(n, 0)); for (int i = 0; i < n; ++i) A[i][i] = 1; // 積の単位元(和の単位元は?) return A; } // O( n^2 ) arr mul(const matrix &A, const arr &x) { arr y(A.size(), 0); for (int i = 0; i < A.size(); ++i) for (int j = 0; j < A[0].size(); ++j) y[i] += A[i][j] * x[j]; // 加群の積と和の演算子 return y; } // O( n^3 ) matrix mul(const matrix &A, const matrix &B) { matrix C(A.size(), arr(B[0].size(), 0)); for (int i = 0; i < C.size(); ++i) for (int j = 0; j < C[i].size(); ++j) for (int k = 0; k < A[i].size(); ++k) C[i][j] += A[i][k] * B[k][j]; // 加群の積と和の演算子 return C; } // O( n^3 log e ) matrix pow(const matrix &A, int e) { return e == 0 ? identity(A.size()) : e % 2 == 0 ? pow(mul(A, A), e/2) : mul(A, pow(A, e-1)); } // O( n ) number inner_product(const arr &a, const arr &b) { number ans = 0; for (int i = 0; i < a.size(); ++i) ans += a[i] * b[i]; return ans; } // O( n ) arr outer_product(const arr &a, const arr &b) { arr ret(3); ret[0] = a[1] * b[2] - a[2] * b[1]; ret[1] = a[2] * b[0] - a[0] * b[2]; ret[2] = a[0] * b[1] - a[1] * b[0]; return ret; } // O( n^3 ) number det(matrix A) { const int n = A.size(); number D = 1; for (int i = 0; i < n; ++i) { int pivot = i; for (int j = i+1; j < n; ++j) if (abs(A[j][i]) > abs(A[pivot][i])) pivot = j; swap(A[pivot], A[i]); D *= A[i][i] * (i != pivot ? -1 : 1); if (abs(A[i][i]) < eps) break; for(int j = i+1; j < n; ++j) for(int k = n-1; k >= i; --k) A[j][k] -= A[i][k] * A[j][i] / A[i][i]; } return D; } // O(n) number tr(const matrix &A) { number ans = 0; for (int i = 0; i < A.size(); ++i) ans += A[i][i]; return ans; } // O( n^3 ). int rank(matrix A) { const int n = A.size(), m = A[0].size(); int r = 0; for (int i = 0; r < n && i < m; ++i) { int pivot = r; for (int j = r+1; j < n; ++j) if (abs(A[j][i]) > abs(A[pivot][i])) pivot = j; swap(A[pivot], A[r]); if (abs(A[r][i]) < eps) continue; for (int k = m-1; k >= i; --k) A[r][k] /= A[r][i]; for(int j = r+1; j < n; ++j) for(int k = i; k < m; ++k) A[j][k] -= A[r][k] * A[j][i]; ++r; } return r; } struct LUinfo { vector value; vector index; }; // O( n^3 ), Gaussian forward elimination LUinfo LU_decomposition(matrix A) { const int n = A.size(); LUinfo data; for (int i = 0; i < n; ++i) { int pivot = i; for (int j = i+1; j < n; ++j) if (abs(A[j][i]) > abs(A[pivot][i])) pivot = j; swap(A[pivot], A[i]); data.index.push_back(pivot); // if A[i][i] == 0, LU decomposition failed. for(int j = i+1; j < n; ++j) { A[j][i] /= A[i][i]; for(int k = i+1; k < n; ++k) A[j][k] -= A[i][k] * A[j][i]; data.value.push_back(A[j][i]); } } for(int i = n-1; i >= 0; --i) { for(int j = i+1; j < n; ++j) data.value.push_back(A[i][j]); data.value.push_back(A[i][i]); } return data; } // O( n^2 ) Gaussian backward substitution arr LU_backsubstitution(const LUinfo &data, arr b) { const int n = b.size(); int k = 0; for (int i = 0; i < n; ++i){ swap(b[data.index[i]], b[i]); for(int j = i+1; j < n; ++j) b[j] -= b[i] * data.value[k++]; } for (int i = n-1; i >= 0; --i) { for (int j = i+1; j < n; ++j) b[i] -= b[j] * data.value[k++]; b[i] /= data.value[k++]; } return b; } // reduce Hessenberg form (inplace). // O ( n^3 ) void hessenberg(matrix &A) { const int n = A.size(); for (int k = 1; k <= n-2; ++k) { arr u(n); for (int i = k; i < n; ++i) u[i] = A[i][k-1]; number ss = 0; for (int i = k+1; i < n; ++i) ss += u[i] * u[i]; if (abs(ss) <= 0.0) continue; number s = sqrt( ss + u[k]*u[k] ); if (u[k] > 0.0) s = -s; u[k] -= s; number uu = sqrt( ss + u[k]*u[k] ); for (int i = k; i < n; ++i) u[i] /= uu; arr f(n), g(n); for (int i = 0; i < n; ++i) for (int j = k; j < n; ++j) f[i] += A[i][j] * u[j], g[i] += A[j][i] * u[j]; number gamma = inner_product(u, g); for (int i = 0; i < n; ++i) f[i] -= gamma * u[i], g[i] -= gamma * u[i]; for (int i = 0; i < n; ++i) for (int j = 0; j < n; ++j) A[i][j] = A[i][j] - 2*u[i]*g[j] - 2*f[i]*u[j]; } } // find all eigenvalues using Hessenberg-QR Method // O( n^3 + M n^2 ) where M is the number of iterations. vector eigenvalues(matrix A) { const int n = A.size(); hessenberg(A); vector s(n), c(n); for (int m = n; m >= 2; ) { if (abs(A[m-1][m-2]) < eps) { --m; continue; } number shift = A[m-1][m-1]; for (int i = 0; i < m; ++i) A[i][i] -= shift; for (int k = 0; k < m-1; ++k) { number a = A[k][k], b = A[k+1][k], r = sqrt(a*a + b*b); s[k] = r == 0.0 ? 0.0 : b/r, c[k] = r == 0.0 ? 0.0 : a/r; for (int j = k; j < m; ++j) { number x = A[k][j], y = A[k+1][j]; A[ k ][j] = c[k] * x + s[k] * y; A[k+1][j] = -s[k] * x + c[k] * y; } } for (int k = 0; k < m-1; ++k) { for (int i = 0; i <= k+1; ++i) { number x = A[i][k], y = A[i][k+1]; A[i][ k ] = c[k] * x + s[k] * y; A[i][k+1] = -s[k] * x + c[k] * y; } } for (int i = 0; i < m; ++i) A[i][i] += shift; } vector lambda; for (int i = 0; i < n; ++i) lambda.push_back( A[i][i] ); return lambda; } // find the corresponding eigenvector from the eigenvalue. // O ( n^3 + M n^2 ) where M is the number of iterations. arr eigenvector(matrix A, number lambda) { const int n = A.size(); arr y(n); y[0] = 1; for (int i = 0; i < n; ++i) A[i][i] -= lambda; LUinfo data = LU_decomposition(A); number mu, v2, v2s; do { arr v = LU_backsubstitution(data, y); // A v = y mu = inner_product(v, y); v2 = inner_product(v, v); v2s = sqrt(v2); for (int j = 0; j < n; ++j) y[j] = v[j] / v2s; } while (abs(1.0-mu*mu/v2) > eps); return y; } int main(void) { vector s; rep(i, 3) { arr tmp(3); cin >> tmp[0] >> tmp[1] >> tmp[2]; s.pb(tmp); } vector xyz(3, arr(3)); rep(i, 3) { rep(j, 3) xyz[0][j] = s[1][j] - s[0][j]; rep(j, 3) xyz[1][j] = s[2][j] - s[0][j]; xyz[2] = outer_product(xyz[0], xyz[1]); } arr a(3); cin >> a[0] >> a[1] >> a[2]; rep(i, 3) a[i] -= s[0][i]; LUinfo lu = LU_decomposition(xyz); arr ret = LU_backsubstitution(lu, a); if (ret[0] >= 0 && ret[1] > 0 && ret[0] + ret[1] <= 1) { cout << "YES" << endl; } else { cout << "NO" << endl; } return 0; }