#include <iostream>
#include <iomanip>
#include <complex>
#include <vector>
#include <algorithm>
#include <cmath>
#include <array>
using namespace std;

const double EPS = 1e-10;
const double INF = 1e12;
#define EQ(n,m) (abs((n)-(m)) < EPS)
#define X real()
#define Y imag()

typedef complex<double> P;
typedef vector<P> VP;
struct L : array<P, 2>{
    L(const P& a, const P& b){ at(0)=a; at(1)=b; }
    L(){}
};
struct C{
    P p;
    double r;
    C(const P& p, const double& r) : p(p), r(r) {}
    C(){}
};

double dot(P a, P b){
    return (conj(a)*b).X;
}
double cross(P a, P b){
    return (conj(a)*b).Y;
}
int ccw(P a, P b, P c){
    b -= a;
    c -= a;
    if(cross(b,c) > EPS) return +1; //ccw
    if(cross(b,c) < -EPS) return -1; //cw
    if(dot(b,c) < -EPS) return +2; //c-a-b
    if(abs(c)-abs(b) > EPS) return -2; //a-b-c
    return 0; //a-c-b
}
P unit(const P &p){
    return p/abs(p);
}
bool intersectSS(const L& a, const L& b){
    return ( ccw(a[0],a[1],b[0]) *ccw(a[0],a[1],b[1]) <= 0 ) &&
        ( ccw(b[0],b[1],a[0]) *ccw(b[0],b[1],a[1]) <= 0 );
}
bool intersectSP(const L& s, const P &p){
    return abs(cross(s[0]-p, s[1]-p))<EPS  && dot(s[0]-p, s[1]-p)<EPS;
}

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]);
}

double distanceLP(const L &l, const P &p) {
    return abs(cross(l[1]-l[0], p-l[0])) /abs(l[1]-l[0]);
}

VP crosspointCL(const C &c, const L &l){
    VP ret;
    P mid = projection(l, c.p);
    double d = distanceLP(l, c.p);
    if(EQ(d, c.r)){
        ret.push_back(mid);
    }else if(d < c.r){
        double len = sqrt(c.r*c.r -d*d);
        ret.push_back(mid +len*unit(l[1]-l[0]));
        ret.push_back(mid -len*unit(l[1]-l[0]));
    }
    return ret;
}
VP crosspointCS(const C &c, const L &s){
    VP ret;
    VP cp = crosspointCL(c,s);
    for(int i=0; i<(int)cp.size(); i++){
        if(intersectSP(s, cp[i])){
            ret.push_back(cp[i]);
        }
    }
    return ret;
}

int in_poly(const P &p, const VP &poly){
    int n = poly.size();
    int ret = -1;
    for(int i=0; i<n; i++){
        P a = poly[i]-p;
        P b = poly[(i+1)%n]-p;
        if(a.Y > b.Y) swap(a,b);
        if(intersectSP(L(a,b), P(0,0))) return 0;
        if(a.Y<=0 && b.Y>0 && cross(a,b)<0) ret = -ret;
    }
    return ret;
}

int main(){
    while(1){
        int x,y;
        VP tri(3);
        cin >> x >> y;
        if(x==0 && y==0) break;
        tri[0] = P(x, y);
        for(int i=1; i<3; i++){
            cin >> x >> y;
            tri[i] = P(x, y);
        }
        if(ccw(tri[0], tri[1], tri[2]) != 1){
            swap(tri[1], tri[2]);
        }

        C c;
        cin >> x >> y >> c.r;
        c.p = P(x, y);
        for(int i=0; i<3; i++){
            tri[i] -= c.p;
        }
        c.p = P(0, 0);

        if(abs(tri[0]) < c.r+EPS && abs(tri[1]) < c.r+EPS && abs(tri[2]) < c.r+EPS){
            cout << "b" << endl;
        }else if(in_poly(c.p, tri) == 1 &&
                 distanceLP(L(tri[0], tri[1]), c.p) +EPS > c.r &&
                 distanceLP(L(tri[1], tri[2]), c.p) +EPS > c.r &&
                 distanceLP(L(tri[2], tri[0]), c.p) +EPS > c.r){
            cout << "a" << endl;
        }else if((crosspointCS(c, L(tri[0], tri[1]))).empty() &&
                 (crosspointCS(c, L(tri[1], tri[2]))).empty() &&
                 (crosspointCS(c, L(tri[2], tri[0]))).empty()){
            cout << "d" << endl;
        }else{
            cout << "c" << endl;
        }
    }
    return 0;
}



Measured memory_bytes: 3284