Language: C++

#include <bits/stdc++.h>
using namespace std;
using ll = long long;
#define rep(i,n) for(int (i)=0;(i)<(int)(n);++(i))
#define all(x) (x).begin(),(x).end()
#define pb push_back
#define fi first
#define se second
#define dbg(x) cout<<#x" = "<<((x))<<endl
template<class T,class U> ostream& operator<<(ostream& o, const pair<T,U> &p){o<<"("<<p.fi<<","<<p.se<<")";return o;}
template<class T> ostream& operator<<(ostream& o, const vector<T> &v){o<<"[";for(T t:v){o<<t<<",";}o<<"]";return o;}
 
/* 基本要素 */
typedef complex<double> Point;
typedef pair<Point, Point> Line;
typedef vector<Point> VP;
const double EPS = 1e-6; // 許容誤差^2
const double INF = 1e9;
#define X real()
#define Y imag()
// #define LE(n,m) ((n) < (m) + EPS)
#define LE(n,m) ((n) - (m) < EPS)
// #define GE(n,m) ((n) + EPS > (m))
#define GE(n,m) (EPS > (m) - (n))
#define EQ(n,m) (abs((n)-(m)) < EPS)
 
// 内積　dot(a,b) = |a||b|cosθ
double dot(Point a, Point b) {
  return a.X*b.X + a.Y*b.Y;
}
 
// 外積　cross(a,b) = |a||b|sinθ
double cross(Point a, Point b) {
  return a.X*b.Y - a.Y*b.X;
}
 
// 点の進行方向
int ccw(Point a, Point b, Point c) {
  b -= a;  c -= a;
  if (cross(b,c) >  EPS) return +1;  // counter clockwise
  if (cross(b,c) < -EPS) return -1;  // clockwise
  if (dot(b,c)   < -EPS) return +2;  // c--a--b on line
  if (norm(b) < norm(c)) return -2;  // a--b--c on line or a==b
  return 0;                          // a--c--b on line or a==c or b==c
}
 
// 点pの直線aへの射影点を返す
Point proj(Point a1, Point a2, Point p) {
  return a1 + dot(a2-a1, p-a1)/norm(a2-a1) * (a2-a1);
}
 
VP crosspointLC(Point a1, Point a2, Point c, double r) {
  VP ps;
  Point ft = proj(a1, a2, c);
  if(!GE(r*r,norm(ft-c))) return ps;
  Point dir = sqrt(max(r*r - norm(ft-c), 0.0)) / abs(a2-a1) * (a2-a1);
  ps.push_back(ft + dir);
  if (!EQ(r*r, norm(ft-c))) ps.push_back(ft - dir);
  return ps;
}
 
// 2円の共通接線。返される各直線に含まれる頂点は円との接点となる
vector<Line> tangentLines(Point a, double ar, Point b, double br) {
  if(ar > br){
    swap(ar,br);
    swap(a,b);
  }

  vector<Line> ls;
  double d = abs(b-a);
  double y = br-ar;
  double theta = asin(y/d);
  double phi = arg(b-a);
  for(int sign : {-1,1}){
    ls.push_back(Line(
      a+polar(ar,phi+sign*(theta+PI/2)), 
      b+polar(br,phi+sign*(theta+PI/2))));
  }
  y = br+ar;
  theta = asin(y/d);
  phi = arg(b-a);
  for(int sign : {-1,1}){
    ls.push_back(Line(
      a+polar(ar,phi+sign*(theta-PI/2)), 
      b+polar(br,phi+sign*(theta-PI/2)+PI)));
  }

  // rep (i,2) {
  //   double sin = (ar - (1-i*2)*br) / d;
  //   if (!LE(sin*sin, 1)) break;
  //   double cos = sqrt(max(1 - sin*sin, 0.0));
  //   rep (j,2) {
  //     Point n = (b-a) * Point(sin, (1-j*2)*cos) / d;
  //     assert(EQ(abs(n),1.0));
  //     ls.push_back(Line(a + ar*n, b + (1-i*2)*br*n));
  //     if (cos < EPS) break;  // 重複する接線を無視（重複していいならこの行不要）
  //   }
  // }
  return ls;
}

const int N = 16;
int dp[1<<N];
int m[N][N][4];
 
int main(){
    int n;
    cin >>n;
 
    vector<Point> c(n);
    vector<int> r(n);
    rep(i,n){
        int x,y;
        cin >>x >>y >>r[i];
        c[i] = Point(x,y);
    }
 
    rep(i,n)rep(j,i){
        vector<Line> seg = tangentLines(c[i],r[i],c[j],r[j]);
        int S = seg.size();
        assert(S==4);
        rep(k,S){
            Line l = seg[k];
            VP tmp = crosspointLC(l.fi,l.se,c[0],r[0]);
            if(tmp.size()==0) continue;
 
            Point p = tmp[0];
            if(ccw(p,l.fi,l.se)==2) continue;
 
            int val = (1<<i)|(1<<j)|1;
            for(int x=1; x<n; ++x)if(x!=i && x!=j){
                VP ump = crosspointLC(p,l.fi,c[x],r[x]);
                if(ump.size()==0) continue;
 
                Point cc = ump[0];
                if(ccw(p,l.fi,cc)!=2) val |= (1<<x);
            }
 
            m[i][j][k] = m[j][i][k] = val;
        }
    }
 
    fill(dp,dp+(1<<N),10101);
    dp[1] = 0;
    rep(mask,1<<n){
        rep(i,n)rep(j,n)rep(k,4){
            int nmask = (mask|m[i][j][k]);
            dp[nmask] = min(dp[nmask], dp[mask]+1);
        }
    }
 
    cout << dp[(1<<n)-1] << endl;
    return 0;
}


Compiler output:
a.cc: In function 'std::vector<std::pair<std::complex<double>, std::complex<double> > > tangentLines(Point, double, Point, double)':
a.cc:76:34: error: 'PI' was not declared in this scope
   76 |       a+polar(ar,phi+sign*(theta+PI/2)),
      |                                  ^~
a.cc:84:34: error: 'PI' was not declared in this scope
   84 |       a+polar(ar,phi+sign*(theta-PI/2)),
      |                                  ^~