#include<string>
#include<vector>
#include<iostream>
#include<algorithm>
#include<cstdio>
#include<cstdlib>
#include<sstream>
#include<functional>
#include<map>
#include<set>
#include<cmath>

#define REP(i,s,e) for(int i=int(s);i<int(e);i++)
#define rep(i,n) for(int i=0;i<int(n);i++)

#define EPS 1e-9

using namespace std;

bool f(double x1,double y1,double x2,double y2,double x4,double y4,double r){

double p=(y2-y1)/(x2-x1);
double q=y1-y4-p*(x1-x4);
double d=p*p*q*q-(1+p*p)*(q*q-r*r);

if(x1!=x2){
if(d>=0 && (-p*q+sqrt(d))/(1+p*p)+EPS<max(x2-x4,x1-x4) && (-p*q-sqrt(d))/(1+p*p)>min(x2-x4,x1-x4)+EPS)
return true;
else return false;
}
else if(abs(x1-x4)<=r && max(y1-y4,y2-y4)>sqrt(r*r-(x1-x4)*(x1-x4))+EPS && min(y1-y4,y2-y4)+EPS<-sqrt(r*r-(x1-x4)*(x1-x4)))
return true;
else return false;
}

bool g(double x1,double y1,double x2,double y2,double x3,double y3,double x4,double y4,double r){

double a=sqrt((x1-x4)*(x1-x4)+(y1-y4)*(y1-y4));
double b=(y1-y4)/(x1-x4);
double c=(y2-y3)/(x2-x3);
double x=(y2-y4-c*(x2-x4))/(b-c);
double y=(b*(y2-y4)-b*c*(x2-x4))/(b-c);

if(x1!=x4 && x2!=x3 && b!=c && a>sqrt((x-x1+x4)*(x-x1+x4)+(y-y1+y4)*(y-y1+y4))+EPS)
return true;
else if(x1==x4 && x2!=x3 && abs(y1-y4)>abs(y1-y2+c*(x2-x4))+EPS)
return true;
else if(x1!=x4 && x2==x3 && a>sqrt((x2-x1)*(x2-x1)+(b*(x2-x4)-y1+y4)*(b*(x2-x4)-y1+y4))+EPS)
return true;
else return false;

}

int main(){

double x1,y1,x2,y2,x3,y3,x4,y4,r;

while(true){

cin >> x1 >> y1 >> x2 >> y2 >> x3 >> y3 >> x4 >> y4 >> r;

if(x1==0 && y1==0) break;


double a,b,c;
a=(x1-x4)*(x1-x4)+(y1-y4)*(y1-y4);
b=(x2-x4)*(x2-x4)+(y2-y4)*(y2-y4);
c=(x3-x4)*(x3-x4)+(y3-y4)*(y3-y4);

if(a<r*r && b<r*r && c<r*r)
cout << "b" << endl;
if(!(a<r*r && b<r*r && c<r*r) && !(a>r*r && b>r*r && c>r*r))
cout << "c" << endl;

if(a>r*r && b>r*r && c>r*r){
if(f(x1,y1,x2,y2,x4,y4,r) || f(x2,y2,x3,y3,x4,y4,r) || f(x1,y1,x3,y3,x4,y4,r))
cout << "c" << endl;
else if(g(x1,y1,x2,y2,x3,y3,x4,y4,r) || g(x2,y2,x1,y1,x3,y3,x4,y4,r) || g(x3,y3,x2,y2,x1,y1,x4,y4,r))
cout << "d" << endl;
else cout << "a" << endl;

}
}

	return 0;
}