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<iostream>
#include<cfloat>
#include<cassert>
#include<cmath>
#include<vector>
#include<algorithm>
#include<stdio.h>
using namespace std;
 
#define EPS (1e-10)
#define equals(a, b) (fabs((a) - (b)) < EPS )
#define dle(a, b) (equals(a, b) || a < b )
static const double PI = acos(-1);

class Point{
public:
  double x, y;
  
  Point ( double x = 0, double y = 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; }
  
  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;
  }
};

class Segment{
public:
  Point p1, p2;
  Segment ( Point p1 = Point(0.0,0.0), Point p2 = Point(0.0,0.0)): p1(p1), p2(p2){}
  double abs() { return sqrt(norm());}
  double norm() { return (p2.x - p1.x)*(p2.x - p1.x) + (p2.y - p1.y)*(p2.y - p1.y); }
};
typedef Segment Line;
typedef Point Vector;
typedef vector<Point> Polygon;
double norm( Vector a ){ return a.x*a.x + a.y*a.y; }
double abs( Vector a ){ return sqrt(norm(a)); }
Point polar( double a, double r ){ return Point(cos(r)*a, sin(r)*a);}
double getDistance( Vector a, Vector b ){ return abs(a - b); }
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; }
bool isOrthogonal( Vector a, Vector b){
  return equals(dot(a,b),0.0);
}
bool isOrthogonal(Segment s1,Segment s2){
  return equals( dot(s1.p2-s1.p1 , s2.p2-s2.p1) ,0.0 );
}
bool isParallel(Vector a, Vector b){
  return equals(cross(a,b),0.0);
}
bool isParallel(Point a1,Point a2,Point b1,Point b2){
  return isParallel(a1-a2,b1-b2);
}
bool isParallel(Segment s1,Segment s2){
  return equals(cross(s1.p2 - s1.p1 , s2.p2 - s2.p1),0.0);
}
Point project(Segment s,Point p){
  Vector base = s.p2-s.p1;
  double t = dot(p-s.p1,base)/norm(base);
  return s.p1+base*t;
}
Point reflect(Segment s,Point p){
  return p + (project(s,p)-p)*2.0;
}

static const int COUNTER_CLOCKWISE = 1;
static const int CLOCKWISE = -1;
static const int ONLINE_BACK = 2;
static const int ONLINE_FRONT = -2;
static const int ON_SEGMENT = 0;
 
// EPS can be 0
// need to check for 920, 833, 866
int ccw( Point p0, Point p1, Point p2 ){
    Vector a = p1 - p0;
    Vector b = p2 - p0;
    if ( cross(a, b) > EPS ) return COUNTER_CLOCKWISE;
    if ( cross(a, b) < -EPS ) return CLOCKWISE;
    if ( dot(a, b) < -EPS ) return ONLINE_BACK;
    if ( norm(a) < norm(b) ) return ONLINE_FRONT;
    return ON_SEGMENT;
}

bool isIntersect(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);
}

bool isIntersect(Segment s1,Segment s2){
  return isIntersect(s1.p1 , s1.p2 , s2.p1 , s2.p2);
}

Point getCrossPoint(Segment s1,Segment s2){
  assert(isIntersect(s1,s2));
  Vector base = s2.p2 - s2.p1;
  double d1 = abs(cross(base, s1.p1-s2.p1));
  double d2 = abs(cross(base, s1.p1-s2.p1));
  double t = d1/(d1+d2);
  return s1.p1 + (s1.p2 - s1.p1)*t;
}

double getDistanceLP(Line s,Point p){
  return abs(cross(s.p2-s.p1,p-s.p1)/abs(s.p2-s.p1));
}

double getDistanceSP(Segment s,Point p){
  if(dot(s.p2-s.p1,p-s.p1)<0.0)return abs(p-s.p1);
  if(dot(s.p1-s.p2,p-s.p2)<0.0)return abs(p-s.p2);
  return getDistanceLP(s,p);
}

double getDistance(Segment s1,Segment s2){
  if(isIntersect(s1,s2))return 0.0;
  return min(min(getDistanceSP(s1,s2.p1),getDistanceSP(s1,s2.p2)),
	     min(getDistanceSP(s2,s1.p1),getDistanceSP(s2,s1.p2)));
}

bool isConvex( Point P[4] ){
    int c = ccw(P[0], P[1], P[2]);
    for ( int i = 1; i < 4; i++ ){
      if ( c != ccw(P[i], P[(i+1)%4], P[(i+2)%4] )) return false;
    }
    return true;
}
bool isConvex( vector<Point> P ){
    int c = ccw(P[0], P[1], P[2]);
    for ( int i = 1; i < 4; i++ ){
      if ( c != ccw(P[i], P[(i+1)%4], P[(i+2)%4] )) return false;
    }
    return true;
}

Polygon andrewScan(Polygon s){
  Polygon 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]);
  
  //int n;
  for(int i=2;i<(int)s.size();i++){
    for(int n=u.size();n>=2&&ccw(u[n-2],u[n-1],s[i])!=CLOCKWISE;n--){
      u.pop_back();
    }
    u.push_back(s[i]);
  }
  
  for(int i=s.size()-3;i>=0;i--){
    for(int n=l.size();n>=2&&ccw(l[n-2],l[n-1],s[i])!=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;
}

double S3( double x1, double y1, double x2,double y2, double x3, double y3 ){
 
  double sum = 0;
  sum = ( x1*y2 - x2*y1 ) + ( x2*y3 - x3*y2 ) + ( x3*y1 - x1*y3 );
  sum = abs(sum);
  sum /= 2;
 
  return sum;
}

bool eq(double x,double y){
  double z=x-y;
  if(z<0.000000001)return true;
  else return false;
}
int check(Point O,Point A,Point B,Point C){
  double a = S3(A.x , A.y , B.x , B.y , O.x , O.y);
  double b = S3(B.x , B.y , C.x , C.y , O.x , O.y);
  double c = S3(C.x , C.y , A.x , A.y , O.x , O.y);

  double d = S3(A.x,A.y,B.x,B.y,C.x,C.y);

  if(eq((a+b+c),d)){
    return 1;
  }else{
    return -1;
  }
}

int main(){
  Point a,b,c,o;
  double r;
  double A,B,C;
  while(cin>>a.x>>a.y){
    if(a.x==0.0&&a.y==0.0)break;
    cin>>b.x>>b.y>>c.x>>c.y>>o.x>>o.y>>r;

    A=getDistance(a,o);
    B=getDistance(b,o);
    C=getDistance(c,o);
    if(A<=r&&B<=r&&C<=r){cout<<"b"<<endl;continue;}

    A=getDistanceSP(Segment(a,b),o);
    B=getDistanceSP(Segment(b,c),o);
    C=getDistanceSP(Segment(c,a),o);

    if(A>=r&&B>=r&&C>=r&&check(o,a,b,c)==1){cout<<"a"<<endl;continue;}
    
    if(A>=r&&B>=r&&C>=r&&check(o,a,b,c)==-1){cout<<"d"<<endl;continue;}
    cout<<"c"<<endl;
  }
  return 0;
}