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<string>
#include<vector>
#include<iostream>
#include<algorithm>
#include<cstdio>
#include<cstdlib>
#include<sstream>
#include<functional>
#include<map>
#include<set>
#include<cmath>

#define REP(i,s,e) for(int i=int(s);i<int(e);i++)
#define rep(i,n) for(int i=0;i<int(n);i++)

#define EPS 1e-9

using namespace std;

bool f(double x1,double y1,double x2,double y2,double x4,double y4,double r){

double p=(y2-y1)/(x2-x1);
double q=y1-y4-p*(x1-x4);
double d=p*p*q*q-(1+p*p)*(q*q-r*r);

if(x1!=x2){
if(d>=0 && (-p*q+sqrt(d))/(1+p*p)+EPS<max(x2-x4,x1-x4) && (-p*q-sqrt(d))/(1+p*p)>min(x2-x4,x1-x4)+EPS)
return true;
else return false;
}
else if(abs(x1-x4)<=r && max(y1-y4,y2-y4)>sqrt(r*r-(x1-x4)*(x1-x4))+EPS && min(y1-y4,y2-y4)+EPS<-sqrt(r*r-(x1-x4)*(x1-x4)))
return true;
else return false;
}

bool g(double x1,double y1,double x2,double y2,double x3,double y3,double x4,double y4,double r){

double a=sqrt((x1-x4)*(x1-x4)+(y1-y4)*(y1-y4));
double b=(y1-y4)/(x1-x4);
double c=(y2-y3)/(x2-x3);
double x=(y2-y4-c*(x2-x4))/(b-c);
double y=(b*(y2-y4)-b*c*(x2-x4))/(b-c);

if(x1!=x4 && x2!=x3 && b!=c && a>sqrt((x-x1+x4)*(x-x1+x4)+(y-y1+y4)*(y-y1+y4))+EPS)
return true;
else if(x1==x4 && x2!=x3 && abs(y1-y4)>abs(y1-y2+c*(x2-x4))+EPS)
return true;
else if(x1!=x4 && x2==x3 && a>sqrt((x2-x1)*(x2-x1)+(b*(x2-x4)-y1+y4)*(b*(x2-x4)-y1+y4))+EPS)
return true;
else return false;

}

int main(){

double x1,y1,x2,y2,x3,y3,x4,y4,r;

while(true){

cin >> x1 >> y1 >> x2 >> y2 >> x3 >> y3 >> x4 >> y4 >> r;

if(x1==0 && y1==0) break;


double a,b,c;
a=(x1-x4)*(x1-x4)+(y1-y4)*(y1-y4);
b=(x2-x4)*(x2-x4)+(y2-y4)*(y2-y4);
c=(x3-x4)*(x3-x4)+(y3-y4)*(y3-y4);

if(a<r*r && b<r*r && c<r*r)
cout << "b" << endl;
if(!(a<r*r && b<r*r && c<r*r) && !(a>r*r && b>r*r && c>r*r))
cout << "c" << endl;

if(a>r*r && b>r*r && c>r*r){
if(f(x1,y1,x2,y2,x4,y4,r) || f(x2,y2,x3,y3,x4,y4,r) || f(x1,y1,x3,y3,x4,y4,r))
cout << "c" << endl;
else if(g(x1,y1,x2,y2,x3,y3,x4,y4,r) || g(x2,y2,x1,y1,x3,y3,x4,y4,r) || g(x3,y3,x2,y2,x1,y1,x4,y4,r))
cout << "d" << endl;
else cout << "a" << endl;

}
}

	return 0;
}

C++: #include <bits/stdc++.h>
using namespace std;

typedef complex<double> P;

const double eps = 1e-8;

struct C {
  P p;
  double r;
  C() : p(P(0,0)), r(0) {}
  C(P p, double r) : p(p), r(r) {}
};

double cross(P a, P b) {
  return a.real()*b.imag() - a.imag()*b.real();
}

double dot(P a, P b) {
  return a.real()*b.real() + a.imag()*b.imag();
}

int ccw(P p, P a, P b) {
  a -= p;
  b -= p;
  if(cross(a,b) > eps) return 1;
  if(cross(a,b) < -eps) return -1;
  if(dot(a,b) < -eps) return 2;
  if(norm(b)-norm(a) > eps) return -2;
  return 0;
}

double getDistanceLP(P s1, P s2, P p) {
  return abs(cross(s2-s1,p-s1))/abs(s2-s1);
}

double getDistanceSP(P s1, P s2, P p) {
  if(dot(s2-s1, p-s1) < 0.0) return abs(p-s1);
  if(dot(s1-s2, p-s2) < 0.0) return abs(p-s2);
  return getDistanceLP(s1, s2, p);
}

bool involve(const C &c, const P &s1, const P &s2) {
  double dist = max(abs(c.p - s1), abs(c.p - s2));
  return c.r - dist > -eps;
}

bool involve(const vector<P> &g, const C &c) {
  int n = g.size();
  for(int i = 0; i < n; ++i) {
    P a = g[i], b = g[(i+1)%n];
    double dist = getDistanceLP(a,b,c.p);
    if(cross(b-a, c.p-a) > -eps && dist - c.r > -eps) {
    } else {
      return false;
    }
  }
  return true;
}

bool involve(const C &c, const vector<P> &g) {
  int n = g.size();
  for(int i = 0; i < n; ++i) {
    P a = g[i], b = g[(i+1)%n];
    if(!involve(c, a, b)) return false;
  }
  return true;
}

bool isIntersect(const P &s1, const P &s2, const C &c) {
  if(involve(c, s1, s2)) return false;
  double dist = getDistanceSP(s1, s2, c.p);
  return c.r - dist > eps;
}

bool isIntersect(const vector<P> &g, const C &c) {
  int n = g.size();
  for(int i = 0; i < n; ++i) {
    P a = g[i], b = g[(i+1)%n];
    if(isIntersect(a,b,c)) return true;
  }
  return false;
}

int main() {
  vector<P> g(3);
  C c;
  while(1) {
    for(int i = 0; i < g.size(); ++i) {
      cin >> g[i].real() >> g[i].imag();
      if(g[0].real() == 0 && g[0].imag() == 0) return 0;
    }
    if(cross(g[1]-g[0], g[2]-g[0]) < 0) swap(g[0], g[1]);
    cin >> c.p.real() >> c.p.imag();
    cin >> c.r;
    if(involve(g,c)) cout << "a" << endl;
    else if(involve(c,g)) cout << "b" << endl;
    else if(isIntersect(g,c)) cout << "c" << endl;
    else cout << "d" << endl;
  }
  return 0;
}