#define _CRT_SECURE_NO_WARNINGS
#define _USE_MATH_DEFINES
#include <cfloat>
#include <climits>
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <ctime>
#include <iostream>
#include <memory>
#include <string>

#include <algorithm>
#include <complex>
#include <list>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <stack>
#include <vector>
#include <bitset>

using namespace std;

#ifdef _MSC_VER
#define __typeof__ decltype
#define strtok_r strtok_s
#endif

#define ITER(c)             __typeof__((c).begin())
#define FOREACH(it, c)      for (ITER(c) it=(c).begin(); it != (c).end(); ++it)
#define RITER(c)            __typeof__((c).rbegin())
#define RFOREACH(it, c)     for (RITER(c) it=(c).rbegin(); it != (c).rend(); ++it)
#define REP(i, n)           REPEAT(i, 0, n)
#define RREP(i, n)          RREPEAT(i, 0, n)
#define REPEAT(i, k, n)     for(int i = (k); i < ((k)+(n)); ++i)
#define RREPEAT(i, k, n)    for(int i = (k)+(n)-1; i >= (k); --i)
#define FROMTO(i,f,t)       for(int i = f; i < t; i++)
#define ALL(c)              (c).begin(), (c).end()
#define LLPOW(p1,p2)        ((ll)pow((double)(p1), (int)p2))
#define ESIZEOF(A)          (sizeof(A)/sizeof((A)[0]))
#define CIN_NO_SYNC         do { cin.tie(0); ios_base::sync_with_stdio(false); } while(0)
#define GETSTR(p)           fgets((p), sizeof(p), stdin)
#define CHOMP(p)            do{ char *_q = (p) + strlen(p)-1; if(*_q == '\n' || *_q == '\r') *_q = 0; } while(0)
#define FILL(m,v)           memset(m, v, sizeof(m))
#define mp                  make_pair
#define pb                  push_back
template<class _T> _T MAX(_T p1,_T p2){return (p1>p2)?p1:p2;} 
template<class _T> _T MIN(_T p1,_T p2){return (p1<p2)?p1:p2;} 
template <class _Tv> inline string join(const _Tv &v,string sep=" "){
    ostringstream s;FOREACH(it,v){if(it!=v.begin())s<<sep;s<<*it;};return s.str();
}

typedef long long ll;
typedef unsigned long long ull;
typedef double D;
typedef complex<D> P;
typedef vector<int> VI;
typedef vector<VI> VVI;
#define X real()
#define Y imag()
#define EPS (1e-9)
#define DEQ(p1,p2) (abs((p1)-(p2)) < EPS)

#ifdef _DEBUG
template<class _Tv> inline void _prfe(const char *_n,const _Tv _c,bool _p=false){
    ITER(_c) _it=_c.begin();
    if(_p){cout<<_n<<" = ["<<endl;for(;_it!=_c.end();++_it)cout<<"    "<<*_it<<endl;cout<<"]"<<endl; }
    else{cout<<_n<<" = [ "<<*_it++;for(;_it!=_c.end();++_it)cout<<", "<<*_it;cout<<" ]"<<endl; }
}
#define pf printf
#define pr(n) do{cout<<#n" = "<<(n)<<endl;}while(0)
#define prfe(n) _prfe(#n, n)
#define prfep(n) _prfe(#n, n, true)
#define dbgchk(n) do{if(n)throw;}while(0)
#else
#define pf(...) /* */
#define pr(...) /* */
#define prfe(...) /* */
#define prfep(...) /* */
#define dbgchk(...) /* */
#endif

inline D inprd(const P &a, const P &b){ return (conj(a) * b).X; }
inline D outprd(const P &a, const P &b){ return (conj(a) * b).Y; }
inline D sign(D d){ return (d<0)?-1:d>0?1:0; }

bool is_intriangle(const P &p1, const P &p2, const P &p3, const P &q){
    D s1 = sign(outprd(p2-p1,q-p1)), s2 = sign(outprd(p3-p2,q-p2)), s3 = sign(outprd(p1-p3,q-p3));
    return (s1 == s2 && s2 == s3);
}
inline P circum_center(const P &a, const P &b, const P &c){
    P bc = c - b, ca = a - c, ab = b - a;
    return (a + b - (inprd(bc, ca) / outprd(ab, bc)) * ab * P(0,1)) / 2.0;
}
D distance_ls_p(const P &a, const P &b, const P &c){
    if(inprd(b-a, c-a) < EPS) return abs(c-a);
    if(inprd(a-b, c-b) < EPS) return abs(c-b);
    return abs(outprd(b-a, c-a)) / abs(b-a);
}

string solve(P p1, P p2, P p3, P pc, int r){
    D dr = r;
    if( distance_ls_p(p1,p2,pc) >= dr - EPS &&
        distance_ls_p(p2,p3,pc) >= dr - EPS &&
        distance_ls_p(p3,p1,pc) >= dr - EPS &&
        is_intriangle(p1,p2,p3,pc))
        return "a";

    if( distance_ls_p(p1,p2,pc) > dr &&
        distance_ls_p(p2,p3,pc) > dr &&
        distance_ls_p(p3,p1,pc) > dr &&
        !is_intriangle(p1,p2,p3,pc))
        return "d";

    if(norm(p1-pc) <= r*r && norm(p2-pc) <= r*r && norm(p3-pc) <= r*r) return "b";

    return "c";
}

int main(){
    P p1,p2,p3,pc;
    int r;
    int x, y;
    while(cin >> x >> y && x && y){
        p1 = P(x,y);
        cin >> x >> y;
        p2 = P(x,y);
        cin >> x >> y;
        p3 = P(x,y);
        cin >> x >> y >> r;
        pc = P(x,y);
        cout << solve(p1,p2,p3,pc,r) << endl;
    }
    return 0;
}

Measured memory_bytes: 1216