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 <vector>
#include <algorithm>
#include <string>
#include <map>
#include <set>
#include <queue>
#include <cstring>
#include <deque>
#include <cstdio>
#include <cmath>
#include <complex>
#include <iostream>
#include <iterator>

using namespace std;

#define reep(i,f,t) for(int i=f ; i<int(t) ; ++i)
#define rep(i,n) reep(i, 0, n) 

typedef vector<int> vi;
typedef vector<vi> vvi;
typedef pair<int,int> pii;

typedef vector<double> vd;
typedef vector<vd> vvd;

const double EPS = 1e-8;
const double INF = 1e12;

// ½ÊãÌ_ExNg
typedef complex<double> P;
namespace std{
	bool operator< (const P& a, const P& b){
		return real(a) != real(b) ? real(a) < real(b) : imag(a) < imag(b);
	}
}

// OÏ
double cross(const P& a, const P& b){
	return imag(conj(a)*b);
}

// àÏ
double dot(const P& a, const P& b){
	return real(conj(a)*b);
}

// ¼ü
struct L : public vector<P> {
	L(const P &a, const P &b){
		push_back(a);
		push_back(b);
	}
};

// ½p`
typedef vector<P> G;

// ~
struct C{
	P p;
	double r;
	C(const P& p, double r) : p(p), r(r) {}
};

int ccw(P a, P b, P c)
{
	b -= a;
	c -= a;
	if(cross(b, c) > 0)   return +1; // ½vñè
	if(cross(b, c) < 0)   return -1; // vñè
	if(dot(b, c) < 0)     return +2; // c--a--b
	if(norm(b) < norm(c)) return -2; // a--b--c
	return 0;
}

// üªÆ_Ìð·»è
bool intersectSP(const L& s, const P& p){
	return abs(s[0]-p) + abs(s[1]-p) - abs(s[1]-s[0]) < EPS;
}

// ¼üãÌ_ÌËe
P projection(const L& l, const P& p){
	double t = dot(p-l[0], l[0]-l[1]) / norm(l[0]-l[1]);
	return l[0] + t*(l[0]-l[1]);
}

// üªÆ_Ì£
double distanceSP(const L& s, const P& p){
	const P r = projection(s, p);
	if(intersectSP(s, r)) return abs(r - p);
	return min(abs(s[0]-p), abs(s[1]-p));
}

char detect(const G& tri, const C& c)
{
	bool inner[3];
	rep(i, 3){
		inner[i] = abs(tri[i] - c.p) - EPS <= c.r;
	}
	if(inner[0] && inner[1] && inner[2])
		return 'b';
	if(inner[0] || inner[1] || inner[2])
		return 'c';
	
	rep(i, 3){
		L l(tri[i], tri[(i+1)%3]);
		if(distanceSP(l, c.p) - EPS <= c.r)
			return 'c';
	}
	
	int v = 0;
	rep(i, 3){
		int now = ccw(tri[i], tri[(i+1)%3], c.p);
		if(now * v < 0)
			return 'd';
		v = now;
	}
	
	return 'a';
}

int main()
{
	for(;;){
		G tri(3);
		rep(i, 3){
			scanf("%lf%lf", &tri[i].real(), &tri[i].imag());
			if(tri[i].real() == 0.0)
				break;
		}
		if(fabs(tri[0].real()) < EPS)
			break;
		
		P p;
		double r;
		scanf("%lf%lf%lf", &p.real(), &p.imag(), &r);
		printf("%c\n", detect(tri, C(p, r)));
	}

	return 0;
}

C++: #include<iostream>
#include<complex>
#define EPS 1e-10
using namespace std;

typedef complex<double> Po;


Po p[3],o;

double cross(Po x, Po y){return x.real()*y.imag() - x.imag()*y.real();}

double area(Po a, Po b, Po c){return abs(cross(b-a,c-a))/2;}

bool inter(Po a, Po b, Po c, Po x){
  return (abs(area(a,b,x)+area(b,c,x)+area(c,a,x)-area(a,b,c)) < EPS);
}

double line_dis(Po a, Po b, Po c){return abs(cross(b-a,c-a))/abs(b-a);}

int main(){
  double x[4],y[4],r;

  for(;;){
    cin >> x[0] >> y[0];
    if(!x[0] && !y[0])break;

    for(int i=1;i<4;i++)cin >> x[i] >> y[i];
    cin >> r;

    for(int i=0;i<3;i++)p[i] = Po(x[i],y[i]);
    o = Po(x[3],y[3]);


    bool f[3];
    if(inter(p[0],p[1],p[2],o)){
      for(int i=0;i<3;i++)f[i] = (line_dis(p[i],p[(i+1)%3],o)>r+EPS);
      if(f[0] && f[1] && f[2])cout << "a\n";
      else{
	for(int i=0;i<3;i++)f[i] = (abs(o-p[i])<r+EPS);
	if(f[0] && f[1] && f[2])cout << "b\n";
	else cout << "c\n";
      }
    }else{
      for(int i=0;i<3;i++)f[i] = (line_dis(p[i],p[(i+1)%3],o)<r+EPS);
      if(f[0] || f[1] || f[2])cout << "c\n";
      else cout << "d\n";
    }
  }
}