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++: #include<bits/stdc++.h>
using namespace std;
#define EPS 1e-10
#define equals(a,b) (fabs( (a) - (b) )< EPS )
// c++ 11,14
#define COUNTER_CLOCKWISE 1
#define CLOCKWISE -1
#define ONLINE_BACK 2
#define ONLINE_FRONT -2
#define ON_SEGMENT 0

typedef struct point{
	double x,y;
	point(){};
	point(double x ,double y):x(x),y(y){};
	point operator + (point &p){ return point(x+p.x,y+p.y);	}
	point operator - (point &p){ return point(x-p.x,y-p.y);	}
	point operator * (point &p){ return point(x*p.x-y*p.y,x*p.y+y*p.x) ;}
	point operator * (double a){ return point(x*a,y*a);	}
	point operator / (double a){ return point(x/a,y/a);	}
	
	double abs() { return sqrt(norm()); }
	double norm() { return x*x+y*y; }
	
	bool operator < (const point &p) const { return x!=p.x ? x<p.x : y<p.y; }
	bool operator == (const point &p) const { return fabs(x-p.x)<EPS && fabs(y-p.y)<EPS; }
}point;

double abs(point a){return a.abs();}
double norm(point a){return a.norm();}
typedef struct circle{
	point p;
	double r;
} circle;
typedef complex<double> C;
typedef struct { 
	point s,e;
} line;

C convert(point a){ return C(a.x,a.y); }
point convert( C a){ return point(a.real(),a.imag() );}
double dot(point a,point b){ return a.x*b.x+a.y*b.y ; } //内積　a・b
double cross(point a,point b){ return a.x*b.y - a.y*b.x ; }//外積(z成分)　a×b
point vec(line l){return l.e-l.s;}
line make(point s,point e){
	line res; res.s=s; res.e=e;
	return res;
}
circle make(point p,double r){ circle c;
	c.p = p ;
	c.r = r;
	return c;
}
point make(){ 
	double x,y; cin>>x>>y;
       	return point(x,y);
}
line lmake(){
	point p0=make();
	point p1=make();
	return make(p0,p1);
}
//直交
bool isorthogonal(point a,point b){ return equals(dot(a,b), 0.0); }
bool isorthogonal(line l1,line l2){ return isorthogonal(vec(l1),vec(l2)); }
//平行
bool isparallel(point a,point b){ return equals(cross(a,b),0.0); }
bool isparallel(line l1,line l2){ return isparallel(vec(l1),vec(l2)); }
//射影
point project(line s,point p){
	point base = vec(s);
	double r=dot(p-s.s,base)/base.norm();
	base = base*r;
	return s.s+base;
}
//反射
point reflect(line l,point p){
	point tmp=project(l,p)-p;
	tmp= tmp*2.0;
	return p+tmp;

}
//交差判定

int ccw(point p0,point p1,point p2){
	point a = p1-p0;
	point b = p2-p0;
if(cross(a,b)>EPS) return 1;//counter_clockwise
if(cross(a,b)<-EPS) return -1;//clockwise
if(dot(a,b)<-EPS)return 2;//online_back
if(a.norm()<b.norm() ) return -2;//online_front
return 0;//on_segment
}
// circle_versino
int ccw(circle c0,circle c1){
	double  d = abs(c0.p-c1.p);
	double r0 = c0.r,r1 = c1.r;
	
	//c0in c1
	if(d<r1-r0)return -2;
	//c1 in c0
	if(d<r0-r1)return 2;
	//c0 cross c1
	if(abs(r0-r1)<=d&&d<=r0+r1)return 1;
	//c0 not cross c1
	return 0;

}

// line p1-p2 line p3-p4
bool intersect(point p1,point p2,point p3,point p4){
	return (ccw(p1,p2,p3)*ccw(p1,p2,p4)<=0 && ccw(p3,p4,p1)*ccw(p3,p4,p2)<=0 ); 
}
// line l1,l2
bool intersect(line l1,line l2){ return intersect(l1.s,l1.e,l2.s,l2.e); }


//距離
//point-point
double distance(point a,point b){ return abs(a-b); }
//point-line(直線）
double distance2(line l,point p){
	return abs(cross(vec(l),p-l.s)/abs(l.e-l.s));
}
//point-line(線分)
double distance(line l,point p){
	if( dot(vec(l),p-l.s) <0.0 ) return abs(p-l.s);
	if( dot(l.s-l.e,p-l.e) <0.0 ) return abs(p-l.e);
	return distance2(l,p);
}
//line-line
double distance(line l1,line l2){
	if(intersect(l1,l2))	return 0.0;
	return min(min(distance(l1,l2.s),distance(l1,l2.e) ), min(distance(l2,l1.s),distance(l2,l1.e) ) );
}

//交点
point crosspoint(line l1,line l2){
	point base = vec(l2);
	double d1 = abs(cross(base,l1.s-l2.s) );
	double d2 = abs(cross(base,l1.e-l2.s) );
	double t = d1/(d1+d2);
	point tmp = vec(l1)*t;
	return l1.s+tmp;
}
//面積
double area(vector<point> p,int n){
	double ans=0.0;
	for(int i=0;i<n-2;i++){
	ans+=cross(p[i+2]-p[0],p[i+1]-p[0]);
	}
	return abs(ans)/2;
}
double area(vector<point> p){
	return area(p,p.size());
}
//内包
int contains(vector<point> &g,point p){
int n=g.size();
bool flag=false;
	for(int i=0;i<n;i++){
		point a=g[i]-p, b=g[(i+1)%n]-p;
		if(abs(cross(a,b) )<EPS && dot(a,b) <EPS)return 1;
		if(a.y>b.y) swap(a,b);
		if(a.y<EPS&&EPS<b.y&&cross(a,b)>EPS) flag=!flag;
	}
	return (flag? 2:0);
}
//凸包
vector<point> andrewscan(vector<point> s){
	vector<point> u,l;
	if(s.size() <3) return s;
	sort(s.begin(),s.end());
	u.push_back(s[0]);
	u.push_back(s[1]);
	l.push_back(s[s.size()-1]);
	l.push_back(s[s.size()-2]);

	for(int i=2;i<s.size();i++){
			//凸包の辺上を含めない場合
		for(int n=u.size();n>=2&&ccw(u[n-2],u[n-1],s[i])!=CLOCKWISE;n--)
			//凸包の辺上を含める場合
		//for(int n=u.size();n>=2&&ccw(u[n-2],u[n-1],s[i])==COUNTER_CLOCKWISE;n--)
		{
			u.pop_back();
		}
		u.push_back(s[i]);
	}

	for(int i=s.size()-3;i>=0;i--){
			//凸包の辺上を含めない場合
		for(int n=l.size();n>=2&&ccw(l[n-2],l[n-1],s[i])!=CLOCKWISE;n--)
			//凸包の辺上を含める場合
		//for(int n=l.size();n>=2&&ccw(l[n-2],l[n-1],s[i])==COUNTER_CLOCKWISE;n--)
		{
			l.pop_back();
		}
		l.push_back(s[i]);
	}

	reverse(l.begin(),l.end());
	for(int i=u.size()-2;i>=1;i--)l.push_back(u[i]);
	return l;
}
//直径
double convex_diameter(vector<point> g){
	int n=g.size();
	int is=0,js=0;
	for(int i=1;i<n;i++){
		if(g[is].y < g[i].y)is =i;
		if(g[i].y < g[js].y)js=i;
	}
	double maxd = abs((g[is]-g[js]));
	int i,mi,j,mj;
	i=mi=is;
	j=mj=js;
	do{
		if(cross(g[(i+1)%n]-g[i],g[(j+1)%n]-g[j])>=0)j=(j+1)%n;
		else i=(i+1)%n;
		if(abs(g[i]-g[j])>maxd){
			//cerr<<i<<' '<<j<<' '<<norm(g[i]-g[j])<<endl;
			maxd =abs(g[i]-g[j]);
			mi=i;mj=j;
		}
	}while(i!=is||j!=js);
	return maxd;
}

int main(){
	while(1){
	vector<point> tri(3);
	tri[0] = make();
	if(tri[0].x==0&&tri[0].y==0)break;
	tri[1] = make();
	tri[2] = make();
	vector<line> l(3);
	l[0] = make(tri[0],tri[1]);
	l[1] = make(tri[1],tri[2]);
	l[2] = make(tri[2],tri[0]);

	point p = make();
	double r ; cin>>r;
	circle c = make(p,r);
	// circle in triangle
	if(r<distance(l[0],p)&&r<distance(l[1],p)&&
			r<distance(l[2],p)&&contains(tri,p) ){
	cout<<'a'<<endl;
	continue;
	}
	// triangle in circle 
	if(r>distance(tri[0],p)&&r>distance(tri[1],p)&&r>distance(tri[2],p)){
	cout<<'b'<<endl;
	continue;
	}
	//cross
	
	if(r>distance(tri[0],p)||r>distance(tri[1],p)||r>distance(tri[2],p)){
	cout<<'c'<<endl;
	continue;
	}
	cout<<'d'<<endl;

	}
	return 0;
}