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<cmath>
#include<algorithm>
#include<iostream>
#include<vector>
#include<string>
#include<climits>
#include<cfloat>

using namespace std;

double EPS = 1e-10;
const double PI = acos(-1);

double add(double a, double b){
if(abs(a+b) < EPS * (abs(a)+abs(b)))return 0;
return a+b;
}

struct point{
double x, y;
point(){}
point(double x,double y) : x(x) , y(y){}

point operator + (point p){
return point(add(x,p.x), add(y,p.y));
}

point operator - (point p){
return point(add(x,-p.x), add(y,-p.y));
}

point operator * (double d){
return point(x*d,y*d);
}

point operator / (double d){
return point(x/d,y/d);
}

bool operator == ( const point &p ) const {
return fabs(x-p.x) < EPS && fabs(y-p.y) < EPS;
}
};

struct circle{ point p; double r; };

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

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


double vec_abs(point a){
return sqrt(a.x*a.x+a.y*a.y);
}

double distance_ls_p(point a, point b, point c) {
if ( dot(b-a, c-a) < EPS ) return vec_abs(c-a);
if ( dot(a-b, c-b) < EPS ) return vec_abs(c-b);
return abs(cross(b-a, c-a)) / vec_abs(b-a);
}

double dist(point a,point b){
return sqrt(pow(a.x-b.x,2)+pow(a.y-b.y,2));
}

int is_point_on_line(point a, point b, point c) {
return cross(b-a, c-a)==0.0 &&
(dot(b-a, c-a) > -EPS) &&
(dot(a-b, c-b) > -EPS);
}

int is_intersected_ls(point a1, point a2, point b1, point b2) {

if(cross(a1-a2,b1-b2)==0){
return is_point_on_line(a1,a2,b1) || is_point_on_line(a1,a2,b2)
|| is_point_on_line(b1,b2,a1) || is_point_on_line(b1,b2,a2);
}
else {
return ( cross(a2-a1, b1-a1) * cross(a2-a1, b2-a1) < EPS ) &&
( cross(b2-b1, a1-b1) * cross(b2-b1, a2-b1) < EPS );
}

}

int crossCS(point a, point b, circle c){
double res=distance_ls_p(a,b,c.p);
double ac=dist(a,c.p);
double bc=dist(b,c.p);

if(res>c.r || (ac<c.r && bc<c.r))return 0;
else if(c.r-res<EPS ||(ac<c.r && bc>=c.r)||(ac>=c.r && bc<c.r))return 1;
else return 2;
}

int inside(point p, vector<point> ps, int n){
point a,b;
a=b=p;
b.x=DBL_MAX;
ps.push_back(ps[0]);
double ymx=-DBL_MAX,ymn=DBL_MAX;
for(int i=0;i ymx=max(ymx,ps[i].y);
ymn=min(ymn,ps[i].y);
}
if(a.y<=ymn||a.y>=ymx)return 0;

for(int i=0;i<n;i++){
if(is_point_on_line(ps[i],ps[i+1],p))return 1;
}

int cnt1=0;
for(int i=0;i<n;i++)
if(is_point_on_line(a,b,ps[i]))cnt1++;

int cnt=0;
for(int i=0;i<n;i++)
if(is_intersected_ls(ps[i],ps[i+1],a,b))cnt++;

return (cnt-cnt1)%2;
}

int crossCP(vector<point> t, circle C){

double ans=0;
int tsz=t.size();

for(int i=0;i<tsz;i++){
double tmp=dist(t[i],C.p);
if(tmp>ans)ans=tmp;
}

if(C.r>=ans)return 2;

int res1=inside(C.p,t,tsz);
double res2=DBL_MAX;

t.push_back(t[0]);

for(int i=0;i<tsz;i++)
res2=min(res2,distance_ls_p(t[i],t[i+1],C.p));

if(res1 && C.r<=res2)return 1;
else if(!res1 && C.r else return 3;
}

int main(void){
point p;
vector t;
circle C;
string ans="zabcd";

while(true){
t.clear();
for(int i=0;i<3;i++){
cin >> p.x >> p.y;
if(p.x==0 && p.y==0)return 0;
t.push_back(p);
}
cin >> C.p.x >> C.p.y >> C.r;

int res=crossCP(t,C);
cout << ans[res] << endl;
}
return 0;
}

C++: //12
#include<iostream>
#include<complex>
#include<cmath>
#include<algorithm>

using namespace std;

typedef complex<double> P;

int cr(P a,P b){
  return a.real()*b.imag()-a.imag()*b.real()+.5;
}

int dt(P a,P b){
  return a.real()*b.real()+a.imag()*b.imag()+.5;
}

int main(){
  for(;;){
    int x,y;
    P t[3],c;
    int r;
    for(int i=0;i<3;i++){
      cin>>x>>y;
      if((i|x|y)==0)return 0;
      t[i]=P(x,y);
    }
    cin>>x>>y;
    c=P(x,y);
    cin>>r;
    if(abs(t[0]-c)<=r&&abs(t[1]-c)<=r&&abs(t[2]-c)<=r){
      cout<<'b'<<endl;
    }else{
      double d[3];
      bool f[2]={};
      for(int i=0;i<3;i++){
	P l=t[(i+1)%3]-t[i];
	if(dt(l,c-t[i])>0&&dt(-l,c-t[(i+1)%3])>0){
	  d[i]=fabs(cr(l,c-t[i])/abs(l));
	}else{
	  d[i]=min(abs(c-t[0]),abs(c-t[1]));
	}
	f[cr(l,c-t[i])<0]=true;
      }
      if(f[0]^f[1]&&d[0]>=r&&d[1]>=r&&d[2]>=r){
	cout<<'a'<<endl;
      }else if(d[0]<=r||d[1]<=r||d[2]<=r){
	cout<<'c'<<endl;
      }else{
	cout<<'d'<<endl;
      }
    }
  }
  return 0;
}