Language: C++

#include<complex>
#include<vector>
#include<algorithm>
using namespace std;

const double EPS = 1e-8;
const double INF = 1e12;
typedef complex<double> P;
namespace std {
  bool operator < (const P& a, const P& b) {
    return real(a) != real(b) ? real(a) < real(b) : imag(a) < imag(b);
  }
}
double cross(const P& a, const P& b) {
  return imag(conj(a)*b);
}
double dot(const P& a, const P& b) {
  return real(conj(a)*b);
}

struct L : public vector<P> {
  L(const P &a, const P &b) {
    push_back(a); push_back(b);
  }
};

typedef vector<P> G;

struct C {
  P p; double r;
  C(const P &p, double r) : p(p), r(r) { }
};

int ccw(P a, P b, P c) {
  b -= a; c -= a;
  if (cross(b, c) > 0)   return +1;       // counter clockwise
  if (cross(b, c) < 0)   return -1;       // clockwise
  if (dot(b, c) < 0)     return +2;       // c--a--b on line
  if (norm(b) < norm(c)) return -2;       // a--b--c on line
  return 0;
}


bool intersectLL(const L &l, const L &m) {
  return abs(cross(l[1]-l[0], m[1]-m[0])) > EPS || // non-parallel
         abs(cross(l[1]-l[0], m[0]-l[0])) < EPS;   // same line
}
bool intersectLS(const L &l, const L &s) {
  return cross(l[1]-l[0], s[0]-l[0])*       // s[0] is left of l
         cross(l[1]-l[0], s[1]-l[0]) < EPS; // s[1] is right of l
}
bool intersectLP(const L &l, const P &p) {
  return abs(cross(l[1]-p, l[0]-p)) < EPS;
}
bool intersectSS(const L &s, const L &t) {
  return ccw(s[0],s[1],t[0])*ccw(s[0],s[1],t[1]) <= 0 &&
         ccw(t[0],t[1],s[0])*ccw(t[0],t[1],s[1]) <= 0;
}
bool intersectSP(const L &s, const P &p) {
  return abs(s[0]-p)+abs(s[1]-p)-abs(s[1]-s[0]) < EPS; // triangle inequality
}

P projection(const L &l, const P &p) {
  double t = dot(p-l[0], l[0]-l[1]) / norm(l[0]-l[1]);
  return l[0] + t*(l[0]-l[1]);
}
P reflection(const L &l, const P &p) {
	return p + P(2,0) * (projection(l, p) - p);
}
double distanceLP(const L &l, const P &p) {
  return abs(p - projection(l, p));
}
double distanceLL(const L &l, const L &m) {
  return intersectLL(l, m) ? 0 : distanceLP(l, m[0]);
}
double distanceLS(const L &l, const L &s) {
  if (intersectLS(l, s)) return 0;
  return min(distanceLP(l, s[0]), distanceLP(l, s[1]));
}
double distanceSP(const L &s, const P &p) {
  const P r = projection(s, p);
  if (intersectSP(s, r)) return abs(r - p);
  return min(abs(s[0] - p), abs(s[1] - p));
}
double distanceSS(const L &s, const L &t) {
  if (intersectSS(s, t)) return 0;
  return min(min(distanceSP(s, t[0]), distanceSP(s, t[1])),
             min(distanceSP(t, s[0]), distanceSP(t, s[1])));
}

#define curr(P, i) Pt[i]
#define next(P, i) Pt[(i+1)%Pt.size()]
enum { OUT, ON, IN };
int contains(const G& Pt, const P& p) {
  bool in = false;
  for (int i = 0; i < Pt.size(); ++i) {
    P a = curr(P,i) - p, b = next(P,i) - p;
    if (imag(a) > imag(b)) swap(a, b);
    if (imag(a) <= 0 && 0 < imag(b))
      if (cross(a, b) < 0) in = !in;
    if (cross(a, b) == 0 && dot(a, b) <= 0) return ON;
  }
  return in ? IN : OUT;
}


int main(){
	double x1,y1,x2,y2,x3,y3,cx,cy,r;
	while(cin>>x1>>y1>>x2>>y2>>x3>>y3>>cx>>cy>>r,x1||y1){
		P pA(x1,y1),pB(x2,y2),pC(x3,y3),pO(cx,cy);
		L c(pA,pB),a(pB,pC),b(pC,pA);
		C Circle(pO,r);
		G ABC; ABC.push_back(pA),ABC.push_back(pB),ABC.push_back(pC);
		
		if(r*r+EPS>max(max(norm(pA-pO),norm(pB-pO)),norm(pC-pO))){
			cout<<"b"; goto NEXT;
		}
		
		if(!ccw(pA,pB,pC))swap(ABC[0],ABC[2]);
		bool ct=contains(ABC,pO);
		if(ct&&min(min(distanceSP(a,pO),distanceSP(b,pO)),distanceSP(c,pO))>r-EPS){
			cout<<"a"; goto NEXT;
		}
		
		if(distanceSP(a,pO)<r+EPS||distanceSP(b,pO)<r+EPS||distanceSP(c,pO)<r+EPS){
			cout<<"c"; goto NEXT;
		}
		
		cout<<"d";
		
		NEXT:
		cout<<endl;
	}
	return 0;
}

Compiler output:
a.cc: In function 'int main()':
a.cc:109:15: error: 'cin' was not declared in this scope
  109 |         while(cin>>x1>>y1>>x2>>y2>>x3>>y3>>cx>>cy>>r,x1||y1){
      |               ^~~
a.cc:4:1: note: 'std::cin' is defined in header '<iostream>'; this is probably fixable by adding '#include <iostream>'
    3 | #include<algorithm>
  +++ |+#include <iostream>
    4 | using namespace std;
a.cc:116:25: error: 'cout' was not declared in this scope
  116 |                         cout<<"b"; goto NEXT;
      |                         ^~~~
a.cc:116:25: note: 'std::cout' is defined in header '<iostream>'; this is probably fixable by adding '#include <iostream>'
a.cc:122:25: error: 'cout' was not declared in this scope
  122 |                         cout<<"a"; goto NEXT;
      |                         ^~~~
a.cc:122:25: note: 'std::cout' is defined in header '<iostream>'; this is probably fixable by adding '#include <iostream>'
a.cc:126:25: error: 'cout' was not declared in this scope
  126 |                         cout<<"c"; goto NEXT;
      |                         ^~~~
a.cc:126:25: note: 'std::cout' is defined in header '<iostream>'; this is probably fixable by adding '#include <iostream>'
a.cc:129:17: error: 'cout' was not declared in this scope
  129 |                 cout<<"d";
      |                 ^~~~
a.cc:129:17: note: 'std::cout' is defined in header '<iostream>'; this is probably fixable by adding '#include <iostream>'
a.cc:131:17: error: jump to label 'NEXT'
  131 |                 NEXT:
      |                 ^~~~
a.cc:116:41: note:   from here
  116 |                         cout<<"b"; goto NEXT;
      |                                         ^~~~
a.cc:120:22: note:   crosses initialization of 'bool ct'
  120 |                 bool ct=contains(ABC,pO);
      |                      ^~