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 <algorithm>
#include <complex>
#include <vector>
#include <cmath>

using namespace std;

#define F first
#define S second

#define X real()
#define Y imag()

#define EPS (1e-7)

typedef complex<double> Point;
typedef pair<double, Point> Circle;

bool isPointInCircle(Point p, Circle c) {
  if(abs(p-c.S) < abs(c.F)) return true;
  return false;
}

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;
}

typedef pair<Point, Point> Segment;
typedef Segment Line;

double distanceLP(Line l, Point p) {
  return abs(cross(l.S-l.F, p-l.F)) / abs(l.S-l.F);
}

double distanceSP(Segment s, Point p) {
  Point a = s.F, b = s.S;
  
  if(dot(b-a, p-a) < EPS) return abs(p-a);
  if(dot(a-b, p-b) < EPS) return abs(p-b);
  return distanceLP(s, p);
}

int main() {
  
  while(1) {
    Point points[3];
    Circle circle;
  
    for(int i=0; i<3; i++) {
      double x, y; cin >> x >> y;
      if(x == 0) return 0;
    
      points[i] = Point(x, y);
    }
  
    {
      double x, y; cin >> x >> y;
      circle.S = Point(x, y);
      cin >> circle.F;
    }
  
    bool ok = 1;
    for(int i=0; i<3; i++) {
      ok &= isPointInCircle(points[i], circle);
    }
    if(ok) {
      cout << 'b' << endl;
      continue;
    }
    
    int uncross = 0;
    for(int i=0; i<3; i++) {
      Segment seg = make_pair(points[(i+1)%3], points[i]);
      uncross += circle.F < distanceSP(seg, circle.S) + EPS;
    }
    if(uncross == 3) {
      int flg = 0;
      for(int i=0; i<3; i++) {
        if(dot(points[(i+1)%3]-points[i], circle.S-points[i]) * dot(points[(i+2)%3]-points[(i+1)%3], circle.S-points[(i+1)%3]) > 0) {
          flg ++;
        }
      }
      if(flg == 3) { cout << 'a' << endl; continue; }
    }

    int cross = 0;
    for(int i=0; i<3; i++) {
      Segment seg = make_pair(points[(i+1)%3], points[i]);
      cross += distanceSP(seg, circle.S) + EPS < circle.F;
    }
    if(cross > 0) cout << 'c' << endl;
    else cout << 'd' << endl;
  }
  
  return 0;
}

C++: #include <cstdio>
#include <iostream>
#include <complex>

using namespace std;

#define dump(n) cerr<<"# "<<#n<<"="<<(n)<<endl

const double EPS=1e-6;
const double INFTY=1e12;
typedef complex<double> Point;

struct Line{
	Point pos,dir;
	Line(){}
	Line(Point p,Point d):pos(p),dir(d){}
};

int Signum(double x)
{
	return abs(x)<EPS?0:x>0?1:-1;
}

double Dot(Point a,Point b)
{
	return real(conj(a)*b);
}

double Cross(Point a,Point b)
{
	return imag(conj(a)*b);
}

int CCW(Point a,Point b,Point c)
{
	Point d1=b-a,d2=c-a;
	int sign=Signum(Cross(d1,d2));
	
	if(sign)
		return sign;	// 1:ccw,-1:cw
	if(Dot(d1,d2)<-EPS)
		return -2;		// c-a-b
	if(norm(d1)<norm(d2)-EPS)
		return 2;		// a-b-c
	return 0;			// a-c-b
}

// ツ点ツづ陳シツ静シツづ個仰猟猟」
double DistancePL(Point p,Line l)
{
	return abs(Cross(p-l.pos,l.dir))/abs(l.dir);
}
// ツ点ツづ静シツ閉ェツづ個仰猟猟」
double DistancePS(Point p,Line s)
{
	Point a=p-s.pos,b=s.dir;
	Point c=b*Dot(a,b)/norm(b);
	int pos=CCW(Point(0,0),b,c);
	if(pos==-2 || pos==2)
		return min(abs(a),abs(b-a));
	return abs(a-c);
}

void solve(Point p[],Point c,double r)
{
	{	// triangle in circle
		double dist=0;
		for(int i=0;i<3;i++)
			dist=max(dist,norm(p[i]-c));
		if(dist<=r*r){
			puts("b");
			return;
		}
	}
	
	{	// circle in triangle
		bool res=true;
		int ccw=0;
		for(int i=0;i<3;i++){
			int temp=CCW(p[i],p[(i+1)%3],c);
			if(temp==-2 || temp==2)
				continue;
			res&=temp*ccw>=0;
			ccw=temp;
		}
		if(res){
			double dist=INFTY;
			for(int i=0;i<3;i++)
				dist=min(dist,DistancePL(c,Line(p[i],p[(i+1)%3]-p[i])));
			if(Signum(dist-r)>=0){
				puts("a");
				return;
			}
		}
	}
	
	{	// intersect
		double dist=INFTY;
		for(int i=0;i<3;i++)
			dist=min(dist,DistancePS(c,Line(p[i],p[(i+1)%3]-p[i])));
		if(Signum(dist-r)<=0){
			puts("c");
			return;
		}
	}
	
	puts("d");
}

int main()
{
	for(;;){
		Point p[3],c;
		double r;
		for(int i=0;i<3;i++)
			cin>>real(p[i])>>imag(p[i]);
		cin>>real(c)>>imag(c)>>r;
		if(norm(p[0])==0)
			break;
		
		solve(p,c,r);
	}
	
	return 0;
}