Language: C++

#include<cmath>
#include<algorithm>
#include<iostream>
#include<vector>
#include<string>
#include<climits>
#include<cfloat>

using namespace std;

double EPS = 1e-10;
const double PI = acos(-1);

double add(double a, double b){
  if(abs(a+b) < EPS * (abs(a)+abs(b)))return 0;
  return a+b;
}

struct point{
  double x, y;
  point(){}
  point(double x,double y) : x(x) , y(y){}

  point operator + (point p){
    return point(add(x,p.x), add(y,p.y));
  }

  point operator - (point p){
    return point(add(x,-p.x), add(y,-p.y));
  }

  point operator * (double d){
    return point(x*d,y*d);
  }

  point operator / (double d){
    return point(x/d,y/d);
  }
  
  bool operator == ( const point &p ) const {
    return fabs(x-p.x) < EPS && fabs(y-p.y) < EPS;
  }
};

struct circle{ point p; double r; };

double dot(point a, point b) { return (a.x * b.x + a.y * b.y); }

double cross(point a, point b) { return (a.x * b.y - a.y * b.x); }


double vec_abs(point a){
  return sqrt(a.x*a.x+a.y*a.y);
}

double distance_ls_p(point a, point b, point c) {
  if ( dot(b-a, c-a) < EPS ) return vec_abs(c-a);
  if ( dot(a-b, c-b) < EPS ) return vec_abs(c-b);
  return abs(cross(b-a, c-a)) / vec_abs(b-a);
}

double dist(point a,point b){
  return sqrt(pow(a.x-b.x,2)+pow(a.y-b.y,2));
}

int is_point_on_line(point a, point b, point c) {
  return cross(b-a, c-a)==0.0 &&
         (dot(b-a, c-a) > -EPS) &&
         (dot(a-b, c-b) > -EPS);
}

point intersection_l(point a1, point a2, point b1, point b2) {
  return a1 + (a2 - a1) * (cross(b2 - b1,b1 - a1) / cross(b2 - b1,a2 - a1));
}

int is_intersected_ls(point a1, point a2, point b1, point b2) {

  if(cross(a1-a2,b1-b2)==0){
    return is_point_on_line(a1,a2,b1) || is_point_on_line(a1,a2,b2) 
        || is_point_on_line(b1,b2,a1) || is_point_on_line(b1,b2,a2);
  }
  else {
    point r=intersection_l(a1,a2,b1,b2);
    return is_point_on_line(a1,a2,r) && is_point_on_line(b1,b2,r);
  }

}
int crossCS(point a, point b, circle c){
  double res=distance_ls_p(a,b,c.p);
  double ac=dist(a,c.p);
  double bc=dist(b,c.p);

  if(res>c.r || (ac<c.r && bc<c.r))return 0;
  else if(c.r-res<EPS ||(ac<c.r && bc>=c.r)||(ac>=c.r && bc<c.r))return 1;
  else return 2;
}

int inside(point p, vector<point> ps, int n){
  segment t;
  t.a=t.b=p;
  t.b.x=100001;
  ps.push_back(ps[0]);
  double ymx=-DBL_MAX,ymn=DBL_MAX;
  for(int i=0;i<n;i++){
    ymx=max(ymx,ps[i].y);
    ymn=min(ymn,ps[i].y);
  }
  if(t.a.y<=ymn||t.a.y>=ymx)return 0;

  for(int i=0;i<n;i++){
    if(is_point_on_line(ps[i],ps[i+1],p))return 1;
  }
  
  int cnt1=0;
  for(int i=0;i<n;i++)
    if(is_point_on_line(t.a,t.b,ps[i]))cnt1++;

  int cnt=0;
  for(int i=0;i<n;i++)
    if(is_intersected_ls(ps[i],ps[i+1],t.a,t.b))cnt++;
    
  return (cnt-cnt1)%2;
}

int crossCP(vector<point> t, circle C){

  double ans=0;
  int tsz=t.size();

  for(int i=0;i<tsz;i++){
    double tmp=dist(t[i],C.p);
    if(tmp>ans)ans=tmp;
  }

  if(C.r>=ans)return 2;

  int res1=inside(C.p,t,tsz);
  double res2=DBL_MAX;

  t.push_back(t[0]);

  for(int i=0;i<tsz;i++)
    res2=min(res2,distance_ls_p(t[i],t[i+1],C.p));

  if(res1 && C.r<=res2)return 1;
  else if(!res1 && C.r<res2)return 4;
  else return 3;
}

int main(void){
  point p;
  vector<point> t;
  circle C;
  string ans="zabcd";

  while(true){
    t.clear();
    for(int i=0;i<3;i++){
      cin >> p.x >> p.y;
      if(p.x==0 && p.y==0)return 0;
      t.push_back(p);
    }
    cin >> C.p.x >> C.p.y >> C.r;

    int res=crossCP(t,C);
    cout << ans[res] << endl;
  }
  return 0;
}

Compiler output:
a.cc: In function 'int inside(point, std::vector<point>, int)':
a.cc:99:3: error: 'segment' was not declared in this scope
   99 |   segment t;
      |   ^~~~~~~
a.cc:100:3: error: 't' was not declared in this scope
  100 |   t.a=t.b=p;
      |   ^