Go: package main

import (
	"bufio"
	"fmt"
	"math"
	"os"
	"strconv"
	"strings"
)

var scanner = bufio.NewScanner(os.Stdin)

func main() {
	for scanner.Scan() {
		buf0 := strings.Split(scanner.Text(), " ")
		input0 := make([]int, 2)
		for i := range input0 {
			input0[i], _ = strconv.Atoi(buf0[i])
		}
		if input0[0] == 0 && input0[1] == 0 {
			break
		}
		a, b := input0[0], input0[1]
		scanner.Scan()
		buf1 := strings.Split(scanner.Text(), " ")
		input1 := make([]int, 2)
		for i := range input1 {
			input1[i], _ = strconv.Atoi(buf1[i])
		}
		c, d := input1[0], input1[1]
		scanner.Scan()
		buf2 := strings.Split(scanner.Text(), " ")
		input2 := make([]int, 2)
		for i := range input2 {
			input2[i], _ = strconv.Atoi(buf2[i])
		}
		e, f := input2[0], input2[1]
		scanner.Scan()
		buf3 := strings.Split(scanner.Text(), " ")
		input3 := make([]int, 2)
		for i := range input3 {
			input3[i], _ = strconv.Atoi(buf3[i])
		}
		g, h := input3[0], input3[1]
		scanner.Scan()
		r, _ := strconv.Atoi(scanner.Text())
		if (a-g)*(a-g)+(b-h)*(b-h) <= r*r && (c-g)*(c-g)+(d-h)*(d-h) <= r*r && (e-g)*(e-g)+(f-h)*(f-h) <= r*r {
			fmt.Println("b")
		} else if dist(a, b, c, d, g, h) > float64(r)-0.000001 && dist(a, b, e, f, g, h) > float64(r)-0.000001 && dist(e, f, c, d, g, h) > float64(r)-0.000001 && inclusive(a, b, c, d, e, f, g, h) {
			fmt.Println("a")
		} else if CL_intersect(a, b, c, d, g, h, r) || CL_intersect(a, b, e, f, g, h, r) || CL_intersect(e, f, c, d, g, h, r) {
			fmt.Println("c")
		} else {
			fmt.Println("d")
		}
	}
}

func CL_intersect(ax, ay, bx, by, cx, cy, r int) bool {
	acx := cx - ax
	acy := cy - ay
	bcx := cx - bx
	bcy := cy - by
	abx := bx - ax
	aby := by - ay
	if acx*acx+acy*acy >= r*r || bcx*bcx+bcy*bcy >= r*r {
		if abx*acx+aby*acy < 0 {
			if acx*acx+acy*acy <= r*r {
				return true
			}
		} else {
			if abx*acx+aby*acy > abx*abx+aby*aby {
				if bcx*bcx+bcy*bcy <= r*r {
					return true
				}
			} else if acx*acx+acy*acy-(abx*acx+aby*acy)*(abx*acx+aby*acy)/(abx*abx+aby*aby) <= r*r {
				return true
			}
		}
	}
	return false
}

func inclusive(ax, ay, bx, by, cx, cy, px, py int) bool {
	ok1 := true
	ok2 := true
	if (bx-ax)*(py-ay)-(px-ax)*(by-ay) < 0 {
		ok1 = false
	}
	if (bx-ax)*(py-ay)-(px-ax)*(by-ay) > 0 {
		ok2 = false
	}
	if (cx-bx)*(py-by)-(px-bx)*(cy-by) < 0 {
		ok1 = false
	}
	if (cx-bx)*(py-by)-(px-bx)*(cy-by) > 0 {
		ok2 = false
	}
	if (ax-cx)*(py-cy)-(px-cx)*(ay-cy) < 0 {
		ok1 = false
	}
	if (ax-cx)*(py-cy)-(px-cx)*(ay-cy) > 0 {
		ok2 = false
	}
	return ok1 || ok2
}

func dist(ax, ay, bx, by, px, py int) float64 {
	dx := float64(bx - ax)
	dy := float64(by - ay)
	A := float64(dx*dx + dy*dy)
	B := dx*float64(ax-px) + dy*float64(ay-py)
	t := -B / A
	if t < 0 {
		t = 0
	}
	if t > 1 {
		t = 1
	}
	tx := float64(ax) + dx*t
	ty := float64(ay) + dy*t
	return math.Sqrt((float64(px)-tx)*(float64(px)-tx) + (float64(py)-ty)*(float64(py)-ty))
}

C++: /**************** 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;
}

Polygon ConvexHull(vector<Point> ps){
	int n=ps.size();
	sort(ps.begin(),ps.end());

	vector<Point> ch_u(n);
	int n_u=0;
	for(int i=n-1;i>=0;i--){
		ch_u[n_u++]=ps[i];
		while(n_u>=3 && ccw(ch_u[n_u-3],ch_u[n_u-2],ch_u[n_u-1])!=CCW){
			ch_u[n_u-2]=ch_u[n_u-1];
			n_u--;
		}
	}

	vector<Point> ch_l(n);
	int n_l=0;
	for(int i=0;i<n;i++){
		ch_l[n_l++]=ps[i];
		while(n_l>=3 && ccw(ch_l[n_l-3],ch_l[n_l-2],ch_l[n_l-1])!=CCW){
			ch_l[n_l-2]=ch_l[n_l-1];
			n_l--;
		}
	}

	Polygon ch;
	for(int i=0;i<n_u-1;i++)	ch.pb(ch_u[i]);
	for(int i=0;i<n_l-1;i++)	ch.pb(ch_l[i]);
	return ch;
}
/*
bool parallel(const Line &l,const Line &m){
	return abs(cross(l[1]-l[0],m[1]-m[0]))<EPS;
}

bool orthogonal(const Line &l,const Line &m){
	return abs(dot(l[1]-l[0],m[1]-m[0]))<EPS;
}
*/
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);
}

Line perp_bisector(const Point &a,const Point &b){
	return Line(Point((a.x+a.y+b.x-b.y)/2,(b.x+b.y-a.x+a.y)/2),
				Point((b.x+b.y+a.x-a.y)/2,(a.x+a.y-b.x+b.y)/2));
}

bool intersect(const Line &l,const Line &m,Point *p=NULL){
	// this routine also returns true in case "M is on L", etc,.
	if(abs(cross(l[1]-l[0],m[1]-m[0]))>EPS
	|| abs(cross(l[1]-l[0],m[0]-l[0]))<EPS){
		if(p){
			double a1,b1,c1,a2,b2,c2;
			calc_abc(l,a1,b1,c1);
			calc_abc(m,a2,b2,c2);
			double det=a1*b2-a2*b1;
			if(abs(det)<EPS)	*p=l[0];	// l == m
			else{
				p->x=(b1*c2-b2*c1)/det;
				p->y=(a2*c1-a1*c2)/det;
			}
		}
		return true;
	}
	return false;
}

bool intersect(const Segment &s,const Segment &t,Point *p=NULL){
	if(max(s[0].x,s[1].x)<min(t[0].x,t[1].x)
	|| max(t[0].x,t[1].x)<min(s[0].x,s[1].x)
	|| max(s[0].y,s[1].y)<min(t[0].y,t[1].y)
	|| max(t[0].y,t[1].y)<min(s[0].y,s[1].y))	return false;

	if(ccw(s[0],s[1],t[0])*ccw(s[0],s[1],t[1])<=0
	&& ccw(t[0],t[1],s[0])*ccw(t[0],t[1],s[1])<=0){
		if(p){
			double a1,b1,c1,a2,b2,c2;
			calc_abc(s,a1,b1,c1);
			calc_abc(t,a2,b2,c2);
			double det=a1*b2-a2*b1;
			if(abs(det)<EPS){	// s is parallel to t
				Point q[3]={s[0],s[1],t[0]};
				for(int i=0;i<3;i++){
					if(dot(q[i]-s[0],q[i]-s[1])<EPS && dot(q[i]-t[0],q[i]-t[1])<EPS){
						*p=q[i];
						break;
					}
				}
			}
			else{
				p->x=(b1*c2-b2*c1)/det;
				p->y=(a2*c1-a1*c2)/det;
			}
		}
		return true;
	}
	return false;
}

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;
	}
}

int intersect(const Circle &c1,const Circle &c2,Point *p=NULL,Point *q=NULL){
	double d2=norm2(c1.c-c2.c);
	if(d2<EPS && abs(c1.r-c2.r)<EPS)	return -1;	// c1==c2
	else if((c1.r+c2.r)*(c1.r+c2.r)+EPS<d2 || d2+EPS<(c1.r-c2.r)*(c1.r-c2.r))	return 0;
	else if(abs(c1.r+c2.r-d2)<EPS){
		if(p&&q)	intersect(c1,Segment(c1.c,c2.c),p,q);
		return 1;
	}
	else{
		if(p&&q){
			double d=sqrt(d2);
			double a=(c1.r*c1.r-c2.r*c2.r+d2)/(2*d);
			double h=sqrt(c1.r*c1.r-a*a);
			Point m=c1.c+a/d*(c2.c-c1.c);
			*p=*q=c2.c;
			p->x+=h*(c2.c.y-c1.c.y)/d;
			p->y-=h*(c2.c.x-c1.c.x)/d;
			q->x-=h*(c2.c.y-c1.c.y)/d;
			q->y+=h*(c2.c.x-c1.c.x)/d;
		}
		return 2;
	}
}

/**************** 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(t-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;
}