#include <bits/stdc++.h>
  
using namespace std;
  
#define rep(i,n) REP(i,0,n)
#define REP(i,s,e) for(int i=(s); i<(int)(e); i++)
#define pb push_back
#define all(r) (r).begin(),(r).end()
#define rall(r) (r).rbegin(),(r).rend()
#define fi first
#define se second
  
typedef long long ll;
typedef vector<int> vi;
typedef vector<ll> vl;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;
 
const int INF = 1e9;
const ll LINF = 1e18;
const ll MOD = 1e9 + 7;
double EPS = 1e-8;

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(){};
  L(const P &a, const P &b) {
    push_back(a); push_back(b);
  }
};

typedef vector<P> G;

struct C {
  P p; double r;
  C(){};
  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 + 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])));
}
P crosspoint(const L &l, const L &m) {
  double A = cross(l[1] - l[0], m[1] - m[0]);
  double B = cross(l[1] - l[0], l[1] - m[0]);
  if (abs(A) < EPS && abs(B) < EPS) return m[0]; // same line
  if (abs(A) < EPS) assert(false); // !!!PRECONDITION NOT SATISFIED!!!
  return m[0] + B / A * (m[1] - m[0]);
}

bool PinC(const P& p, const C& c) {
	return abs(p-c.p) < c.r+EPS;
}

bool PinPol(const P& p, const G& pol){
	if(pol.size() < 2) return false;
	if(intersectSP({pol[0], pol[1]}, p)) return true;
	bool f = cross(pol[1]-pol[0], p - pol[0]) > 0;
	for(int i = 1; i < pol.size(); i++) {
		auto& now = pol[i];
		auto& nxt = pol[(i+1) % pol.size()];
		if(intersectSP({nxt, now}, p)) return true;
		if(f^(cross(nxt-now, p-now)>0)) return false;
	}
	return true;
}

//?????¨?????????????????¢???(??????????????£??????????????£?????????)
int pos_S_C(const L&s, const C&c){
	if(abs(distanceSP(s,c.p)-c.r)<EPS){
		return 1;	//??\?????????
	}
	if(distanceSP(s,c.p) > c.r+EPS) {
		return 0;	//????????£????????????
	}
	if(PinC(s[0], c) && PinC(s[1], c)){
		return 2;	//????????????
	}
	if(PinC(s[0], c)||PinC(s[1], c)){
		return 3;	//?????´???????????????
	}
	return 4;	//????????????
}

int main(){
	double a, b;
	while(cin>>a>>b){
		if(a == 0.0 && b == 0.0) break;
		G tri;
		tri.pb({a, b});
		rep(i, 2) {
			cin >> a >> b;
			tri.pb({a, b});
		}
		C c;
		cin >> a>> b;
		c.p = {a, b};
		cin >> a;
		c.r = a;
		if(PinC(tri[0], c)&& PinC(tri[1], c) && PinC(tri[2], c)){
			cout<<"b"<<endl;
			continue;
		}
		bool in = PinPol(c.p, tri), hit = false;
		rep(i, 3){
			L l;
			l.pb(tri[i]);
			l.pb(tri[(i+1)%3]);
			in &= pos_S_C(l, c) <= 1;
			hit |= pos_S_C(l, c) > 0;
		}
		if(in) cout<<"a"<<endl;
		else if(hit) cout<<"c"<<endl;
		else cout<<"d"<<endl;
	}
}