/**************** Geometrical Library ****************/

#include<cmath>
#include<vector>
#include<algorithm>

#define	EPS	1e-9

enum {CCW=1,CW=-1,ON=0};

#define	mp	make_pair
#define	pb	push_back

using namespace std;

class Point{
public:
	double x,y;
	Point(){}
	Point(double xx,double yy):x(xx),y(yy){}
};

class Line:public vector<Point>{
public:
	Line(const Point &a,const Point &b){
		pb(a),pb(b);
	}
};

class Segment:public Line{
public:
	Segment(const Point &a,const Point &b):Line(a,b){}
};

class Polygon:public vector<Point>{};

class Circle{
public:
	Point c;
	double r;
	Circle(){}
	Circle(const Point &cc,double rr):c(cc),r(rr){}
};

Point &operator +=(Point &a,const Point &b){
	a.x+=b.x,a.y+=b.y;
	return a;
}

Point &operator -=(Point &a,const Point &b){
	a.x-=b.x,a.y-=b.y;
	return a;
}

Point &operator *=(Point &a,double c){
	a.x*=c,a.y*=c;
	return a;
}

Point &operator /=(Point &a,double c){
	a.x/=c,a.y/=c;
	return a;
}

Point operator +(const Point &a,const Point &b){
	Point c=a;
	return c+=b;
}

Point operator -(const Point &a,const Point &b){
	Point c=a;
	return c-=b;
}

Point operator *(double c,const Point &a){
	Point b=a;
	return b*=c;
}

Point operator /(const Point &a,double c){
	Point b=a;
	return b/=c;
}

bool operator <(const Point &a,const Point &b){
	return (a.x==b.x)?(a.y<b.y):(a.x<b.x);
}

bool operator >(const Point &a,const Point &b){
	return b<a;
}

double dot(const Point &a,const Point &b){
	return a.x*b.x+a.y*b.y;
}

double cross(const Point &a,const Point &b){
	return a.x*b.y-a.y*b.x;
}

double norm2(const Point &a){
	return dot(a,a);
}

int ccw(const Point &a,Point b,Point c){
	b-=a,c-=a;
	double rotdir=cross(b,c);
	if(rotdir>EPS)		return CCW;
	if(rotdir<-EPS)		return CW;
	return ON;
}

inline void calc_abc(const Line &l,double &a,double &b,double &c){	// l : ax+by+c=0
	a=l[0].y-l[1].y;
	b=l[1].x-l[0].x;
	c=l[0].x*l[1].y-l[1].x*l[0].y;
}

Point perp_foot(const Point &p,const Line &l){
	double a,b,c;
	calc_abc(l,a,b,c);
	return p-(a*p.x+b*p.y+c)/(a*a+b*b)*Point(a,b);
}

int intersect(const Circle &c,const Line &l,Point *p=NULL,Point *q=NULL){
	Point m=perp_foot(c.c,l);
	double d2=norm2(c.c-m);
	if(c.r*c.r+EPS<d2)	return 0;
	else if(abs(c.r*c.r-d2)<EPS){
		if(p&&q)	*p=*q=m;
		return 1;
	}
	else{
		if(p&&q){
			Point v;
			double n0=norm2(l[0]-m),n1=norm2(l[1]-m);
			if(n0<n1)	v=l[1]-m,n0=n1;
			else		v=l[0]-m;
			v*=sqrt((c.r*c.r-d2)/n0);
			*p=m+v,*q=m-v;
		}
		return 2;
	}
}

int intersect(const Circle &c,const Segment &s,Point *p=NULL,Point *q=NULL){
	Point pp,qq;
	int n=intersect(c,Line(s[0],s[1]),&pp,&qq);
	if(n==0)	return 0;
	else if(n==1){
		if(dot(pp-s[0],pp-s[1])<EPS){
			if(p&&q)	*p=*q=pp;
			return 1;
		}
		return 0;
	}
	else{
		bool bp=(dot(pp-s[0],pp-s[1])<EPS);
		bool bq=(dot(qq-s[0],qq-s[1])<EPS);
		if(bp&&bq){
			if(p&&q)	*p=pp,*q=qq;
			return 2;
		}
		if(bp||bq){
			if(p&&q)	*p=(bp?pp:qq);
			return 1;
		}
		return 0;
	}
}

/**************** Library END ****************/

#include<cstdio>

int main(){
	for(Point p,q,r;scanf("%lf%lf%lf%lf%lf%lf",&p.x,&p.y,&q.x,&q.y,&r.x,&r.y),p.x||p.y;){
		Circle c;	scanf("%lf%lf%lf",&c.c.x,&c.c.y,&c.r);

		int cnt=ccw(p,q,c.c)+ccw(q,r,c.c)+ccw(r,p,c.c);
		Point s=perp_foot(c.c,Line(p,q));
		Point t=perp_foot(c.c,Line(q,r));
		Point u=perp_foot(c.c,Line(r,p));
		if((cnt==3*CCW || cnt==3*CW)
		&& c.r*c.r<norm2(s-c.c)+EPS
		&& c.r*c.r<norm2(t-c.c)+EPS
		&& c.r*c.r<norm2(u-c.c)+EPS){
			puts("a");
			continue;
		}

		if(norm2(p-c.c)<c.r*c.r+EPS
		&& norm2(q-c.c)<c.r*c.r+EPS
		&& norm2(r-c.c)<c.r*c.r+EPS){
			puts("b");
			continue;
		}

		if(intersect(c,Segment(p,q))==0
		&& intersect(c,Segment(q,r))==0
		&& intersect(c,Segment(r,p))==0){
			puts("d");
			continue;
		}

		puts("c");
	}

	return 0;
}

Measured memory_bytes: 0