#include <iostream>
#include <algorithm>
#include <vector>
#include <cmath>
using namespace std;
#define DEBUG(x) cerr << #x << " = " << x << endl
#define REP(i,n) for(int i=0;i<(int)(n);++i)
typedef double D;
const D EPS = 1e-8;
const D PI = M_PI;
struct P {
  D x, y;
  P() { }
  P(D x_, D y_) : x(x_), y(y_) { }
};
struct L {
  P a, b;
  L() { }
  L(P a_, P b_) : a(a_), b(b_) { }
};
struct C {
  P p; D r;
  C() { }
  C(P p_, D r_) : p(p_), r(r_) { }
};
P operator +(P a, P b) { return P(a.x + b.x, a.y + b.y); }
P operator -(P a, P b) { return P(a.x - b.x, a.y - b.y); }
P operator *(P a, D b) { return P(a.x * b, a.y * b); }
P operator /(P a, D b) { return P(a.x / b, a.y / b); }
P vec(P st, P en) { return en - st; }
D inp(P a, P b) { return a.x*b.x + a.y*b.y; }
D outp(P a, P b) { return a.x*b.y - a.y*b.x; }
D abs(P p) { return sqrt(inp(p,p)); }
int sgn(D a, D b) {
  if(a < b - EPS) return -1;
  if(a > b + EPS) return +1;
  return 0;
}
P proj(L l, P p) {
  P a = vec(l.a,l.b);
  P b = vec(l.a,p);
  D t = inp(a,b)/abs(a);
  return l.a + (a/abs(a))*t;
}
int ccw(P a, P b, P c) {
  b = vec(a,b); c = vec(a,c);
  if(sgn(outp(b,c),0.0) > 0) return +1;
  if(sgn(outp(b,c),0.0) < 0) return -1;
  if(sgn(inp(b,c),0.0) < 0)  return +2;
  if(sgn(abs(b),abs(c)) < 0) return -2;
  return 0;
}
D dLP(L l, P p) {
    return abs(p - proj(l, p));
}
bool iSP(L s, P p) {
    return ccw(s.a, s.b, p) == 0;
}
D dSP(L s, P p) {
  P r = proj(s, p);
  if(iSP(s, r)) return abs(p - r); // 写像がs上にある
  return min(abs(p - s.a), abs(p - s.b)); // 写像がs上にない
}
int containsGP(vector<P> g, P p) {
  int side = -1;
  for(int i = 0; i < (int)g.size(); i++) {
    if(ccw(g[i], g[(i+1)%g.size()], p) == 0) return 0;
    P a = vec(p, g[i]);
    P b = vec(p, g[(i+1)%g.size()]);
    if(a.y > b.y) swap(a, b);
    if(sgn(a.y, 0.0) <= 0 && sgn(b.y, 0.0) > 0 && sgn(outp(a, b), 0.0) > 0) side *= -1;
  }
  return side;
}
int iCS(C c, L s) {
  if(sgn(abs(c.p - s.a), c.r) <= 0 && sgn(abs(c.p - s.b), c.r) <= 0)
    return -2;
  if(sgn(abs(c.p - s.a), c.r) <= 0 || sgn(abs(c.p - s.b), c.r) <= 0)
    return -1;
  if(sgn(dSP(s, c.p), c.r) < 0)
    return +2;
  if(sgn(dSP(s, c.p), c.r) == 0)
    return +1;
  return 0;
}
int main() {
  while(true) {
    int x, y;
    vector<P> tri;
    cin >> x >> y;
    if(x == 0 && y == 0) break;
    tri.emplace_back(x, y);
    cin >> x >> y;
    tri.emplace_back(x, y);
    cin >> x >> y;
    tri.emplace_back(x, y);
    C cir;
    cin >> cir.p.x >> cir.p.y >> cir.r;
    // cerr << "circle(" << cir.p.x << "," << cir.p.y << "," << cir.r << ");" << endl;
    // cerr << "line(" << tri[0].x << "," << tri[0].y << "," << tri[1].x << "," << tri[1].y << ");" << endl;
    // cerr << "line(" << tri[1].x << "," << tri[1].y << "," << tri[2].x << "," << tri[2].y << ");" << endl;
    // cerr << "line(" << tri[2].x << "," << tri[2].y << "," << tri[0].x << "," << tri[0].y << ");" << endl;
    // b
    if(sgn(abs(vec(cir.p, tri[0])), cir.r) <= 0 &&
       sgn(abs(vec(cir.p, tri[1])), cir.r) <= 0 &&
       sgn(abs(vec(cir.p, tri[2])), cir.r) <= 0) {
      cout << "b" << endl;
      continue;
    }
    double R = min({dSP(L(tri[0], tri[1]), cir.p), dSP(L(tri[1], tri[2]), cir.p), dSP(L(tri[2], tri[0]), cir.p)});
    // a
    if(containsGP(tri, cir.p) >= 0 &&
       sgn(cir.r, R) <= 0) {
      cout << "a" << endl;
      continue;
    }
    if(iCS(cir, L(tri[0], tri[1])) == 0 &&
       iCS(cir, L(tri[1], tri[2])) == 0 &&
       iCS(cir, L(tri[2], tri[0])) == 0) {
      cout << "d" << endl;
      continue;
    }
    cout << "c" << endl;
  }
}

Measured memory_bytes: 1232