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++: //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();
}

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

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[i]),abs(c-t[(i+1)%3]));
	}
	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;
}

C++: #include <iostream>
#include <cmath>

using namespace std;

#define EPS 1e-7

class Point{
public:
  double x,y;
  Point(double x = 0.0,double y = 0.0) : 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 * (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; }
};

typedef Point Vector;
double dot(Vector a,Vector b){
  return a.x*b.x + a.y*b.y;
}
double cross(Vector a,Vector b){
  return a.x*b.y - a.y*b.x;
}

class Segment{
public:
  Vector t,s;
  Segment(Vector t,Vector s) : t(t),s(s) {}
  Point projection(Point &p){
    Vector b = s - t;
    double d = dot(p-t,b) / b.norm();
    return t + b*d;
  }
};

class Circle{
public:
  double x,y,r;
  Circle(double x = 0.0,double y = 0.0,double r = 0.0) : x(x),y(y),r(r) {}
};

Point point[3];
Circle ci;

int ccw(Point p0,Point p1,Point p2){
  Vector a = p1 - p0;
  Vector b = p2 - p0;
  if(cross(a,b) > EPS) return 1;
  if(cross(a,b) < -EPS) return -1;
  if(dot(a,b) < -EPS) return 2;
  if(a.norm() < b.norm()) return -2;
  return 0;
}

bool check(){
  Point cp = Point(ci.x,ci.y);
  int c1 = ccw(point[0],point[1],cp);
  int c2 = ccw(point[1],point[2],cp);
  int c3 = ccw(point[2],point[0],cp);
 
  if(c1 == c2 && c2 == c3) return true;
  return false;
}

typedef Segment Line;
double getDistanceLP(Line l,Point p){
  return abs(cross(l.s-l.t,p-l.t)/(l.s-l.t).abs());
}

double getDistanceSP(Segment seg,Point p){
  if(dot(seg.s-seg.t,p-seg.t) < 0.0) return (p-seg.t).abs();
  if(dot(seg.t-seg.s,p-seg.s) < 0.0) return (p-seg.s).abs(); 
  return getDistanceLP(seg,p);
}

int getStateCS(Circle c,Segment seg){
  Point cp = Point(c.x,c.y);
  Point p = seg.projection(cp);
  Point pp = p - cp;
  Vector a = seg.t - cp;
  Vector b = seg.s - cp;
  if(check() && a.abs() > c.r && b.abs() > c.r){
    if(pp.abs()+EPS > c.r && ccw(seg.t,seg.s,p)==0){
      return 1;
    }
  }else if(a.abs() < c.r && b.abs() < c.r){
    if(pp.abs()-EPS < c.r && ccw(seg.t,seg.s,p)==0){
      return 2;
    }
  }
  return 3;
}

bool isCross(){
  Point cp = Point(ci.x,ci.y);
  for(int i = 0 ; i < 3 ; i++){
    Point p = point[i];
    Point diff = p - cp;
    if(diff.abs() == ci.r) return true;
  }
  Segment seg[3] = {
    Segment(point[0],point[1]),
    Segment(point[1],point[2]),
    Segment(point[2],point[0])
  };
  
  for(int i = 0 ; i < 3 ; i++){
    if(getDistanceSP(seg[i],cp) < ci.r){
      return true;
    }
  }

  for(int i = 0 ; i < 3 ; i++){
    Segment ns = seg[i];
    Point p1 = ns.t, p2 = ns.s;
    if(p1.abs() < ci.r && p2.abs() > ci.r) return true;
    if(p1.abs() > ci.r && p2.abs() < ci.r) return true;
  }

  return false;
}

void judge(){
  Segment seg[3] = {
    Segment(point[0],point[1]),
    Segment(point[1],point[2]),
    Segment(point[2],point[0])
  };

  int CinT = 0,TinC = 0;
  for(int i = 0 ; i < 3 ; i++){
    int N = getStateCS(ci,seg[i]);
    if(N == 1){
      CinT++;
    }else if(N == 2){
      TinC++;
    }
  }

  if(CinT == 3){
    cout << 'a' << endl;
  }else if(TinC == 3){
    cout << 'b' << endl;
  }else{
    if(isCross()){
      cout << 'c' << endl;
    }else{
      cout << 'd' << endl;
    }
  }
}

int main(){
  while(true){
    cin >> point[0].x >> point[0].y;
    if(!point[0].x && !point[0].y) break;
    for(int i = 0 ; i < 2 ; i++){
      cin >> point[i+1].x >> point[i+1].y;
    }
    cin >> ci.x >> ci.y >> ci.r;
    judge();
  }

  return 0;
}