#include "bits/stdc++.h"
using namespace std;
typedef complex<double> P;

int cr(P a,P b){
  return a.real()*b.imag()-a.imag()*b.real();
}

int dt(P a,P b){
  return a.real()*b.real()+a.imag()*b.imag();
}

int main(){
  for(;;){int x,y; P t[3],c;int r;
    for(int i=0;i<3;i++){cin>>x>>y;if((i|x|y)==0)return 0;t[i]=P(x,y);}
    cin>>x>>y;c=P(x,y);cin>>r;
    if(abs(t[0]-c)<=r&&abs(t[1]-c)<=r&&abs(t[2]-c)<=r){cout<<'b'<<endl;}
    else{double d[3];bool f[2]={};
      for(int i=0;i<3;i++){
        P l=t[(i+1)%3]-t[i];
        if(dt(l,c-t[i])>0&&dt(-l,c-t[(i+1)%3])>0){
        	d[i]=fabs(cr(l,c-t[i])/abs(l));
        	}else{
        		d[i]=min(abs(c-t[i]),abs(c-t[(i+1)%3]));
        	}f[cr(l,c-t[i])<0]=true;
      	  }if(f[0]^f[1]&&d[0]>=r&&d[1]>=r&&d[2]>=r){
      		  cout<<'a'<<endl;
      	  }else if(d[0]<=r||d[1]<=r||d[2]<=r){
      		  cout<<'c'<<endl;
      	  }else{
    	  cout<<'d'<<endl;
      }
    }
  }
  return 0;
}