#include <cstdio>
#include <iostream>
#include <complex>

using namespace std;

#define dump(n) cerr<<"# "<<#n<<"="<<(n)<<endl

const double EPS=1e-9;
const double INFTY=1e12;
typedef complex<double> Point;

struct Line{
	Point pos,dir;
	Line(){}
	Line(Point p,Point d):pos(p),dir(d){}
};

int Signum(double x)
{
	return abs(x)<EPS?0:x>0?1:-1;
}

double Dot(Point a,Point b)
{
	return real(conj(a)*b);
}

double Cross(Point a,Point b)
{
	return imag(conj(a)*b);
}

int CCW(Point a,Point b,Point c)
{
	Point d1=b-a,d2=c-a;
	int sign=Signum(Cross(d1,d2));
	
	if(sign)
		return sign;	// 1:ccw,-1:cw
	if(Dot(d1,d2)<-EPS)
		return -2;		// c-a-b
	if(norm(d1)<norm(d2)-EPS)
		return 2;		// a-b-c
	return 0;			// a-c-b
}

// ツ点ツづ陳シツ静シツづ個仰猟猟」
double DistancePL(Point p,Line l)
{
	return abs(Cross(p-l.pos,l.dir))/abs(l.dir);
}
// ツ点ツづ静シツ閉ェツづ個仰猟猟」
double DistancePS(Point p,Line s)
{
	Point a=p-s.pos,b=s.dir;
	Point c=b*Dot(a,b)/norm(b);
	double pos=abs(c)/abs(b);
	if(-EPS<pos && pos<1+EPS)
		return abs(a-c);
	else
		return min(abs(a),abs(b-a));
}

void solve(Point p[],Point c,int r)
{
	{	// triangle in circle
		double dist=0;
		for(int i=0;i<3;i++)
			dist=max(dist,norm(p[i]-c));
		if(dist<=r*r){
			puts("b");
			return;
		}
	}
	
	{	// circle in triangle
		bool res=true;
		int ccw=0;
		for(int i=0;i<3;i++){
			int temp=CCW(p[i],p[(i+1)%3],c);
			if(abs(temp)==2)
				continue;
			res&=temp*ccw>=0;
			ccw=temp;
		}
		if(res){
			double dist=INFTY;
			for(int i=0;i<3;i++)
				dist=min(dist,DistancePL(c,Line(p[i],p[(i+1)%3]-p[i])));
			if(Signum(dist-r)>=0){
				puts("a");
				return;
			}
		}
	}
	
	{	// intersect
		double dist=INFTY;
		for(int i=0;i<3;i++)
			dist=min(dist,DistancePS(c,Line(p[i],p[(i+1)%3]-p[i])));
		if(Signum(dist-r)<=0){
			puts("c");
			return;
		}
	}
	
	puts("d");
}

int main()
{
	for(;;){
		Point p[3],c;
		double r;
		for(int i=0;i<3;i++)
			cin>>real(p[i])>>imag(p[i]);
		cin>>real(c)>>imag(c)>>r;
		if(norm(p[0])==0)
			break;
		
		//for(int i=0;i<3;i++)
		//	printf("(%.3f,%.3f) ",real(p[i]),imag(p[i]));
		//printf("(%.3f,%.3f,%.3f)\n",real(c),imag(c),r);
		
		solve(p,c,r);
	}
	
	return 0;
}