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 <sstream>
#include <string>
#include <algorithm>
#include <vector>
#include <map>
#include <set>
#include <bitset>
#include <queue>
#include <stack>
#include <utility>
#include <complex>
#include <cstring>
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <cctype>
using namespace std;
 
#define rep(i,n) for(int i=0; i<n; i++)
#define repa(i,s,e) for(int i=s; i<=e; i++)
#define repd(i,s,e) for(int i=s; i>=e; i--)

typedef pair<int,int> pii;
typedef vector<int> vi;
typedef long long ll;

typedef complex<double> P;

int x, y, r;
P tri[3];
pair<P,double> circle;

double cross(P& e1, P& e2) {
	return e1.real() * e2.imag() - e1.imag() * e2.real();
}

bool inc() {
	P e0 = tri[0] - circle.first;
	P e1 = tri[1] - circle.first;
	P e2 = tri[2] - circle.first;
	double d0 = cross(e0, e1);
	double d1 = cross(e1, e2);
	double d2 = cross(e2, e0);
	if(d0 * d1 >= 0 && d1 * d2 >= 0 && d2 * d0 >= 0) return true;
	return false;
}

double norm(P p) {
	return sqrt(p.real() * p.real() + p.imag() * p.imag());
}

double dist(P p1, P p2, P p) {
	double a = p2.imag() - p1.imag();
	double b = -(p2.real() - p1.real());
	double c = - a * p1.real() - b * p1.imag();
	double nm = a * p.real() + b * p.imag() + c;
	double de = a * a + b * b;
	if(de == 0.0) {
		return norm(p1 - p);
	}
	return abs(nm) / sqrt(de);
}


void solve() {
	bool iscin = inc();
	double R = circle.second;
	vector<double> d(6, 0.0);
	d[0] = dist(tri[0], tri[1], circle.first);
	d[1] = dist(tri[1], tri[2], circle.first);
	d[2] = dist(tri[2], tri[0], circle.first);
	// printf("%f %f %f\n", d[0], d[1], d[2]);
	if(iscin && d[0] > R && d[1] > R && d[2] > R) {
		cout << "a" << endl;
		return;
	}

	d[3] = norm(tri[0] - circle.first);
	d[4] = norm(tri[1] - circle.first);
	d[5] = norm(tri[2] - circle.first);
	// printf("%f %f %f\n", d[3], d[4], d[5]);
	if(d[3] < R && d[4] < R && d[5] < R) {
		cout << "b" << endl;
		return;
	}

	sort(d.begin(), d.end());
	if(d[0] < R) {
		cout << "c" << endl;
		return;
	}

	cout << "d" << endl;
}

int main() {
	while( 1 ) {
		scanf("%d%d", &x, &y);
		if(x == 0 && y == 0) break;
		tri[0] = P(x, y);

		scanf("%d%d", &x, &y);
		tri[1] = P(x, y);
		
		scanf("%d%d", &x, &y);
		tri[2] = P(x, y);
		
		scanf("%d%d%d", &x, &y, &r);
		circle.first = P(x, y);
		circle.second = r;

		solve();
	}
}

C++: #include <iostream>
#include <complex>
#include <vector>
#include <algorithm>
using namespace std;

typedef complex<double> P;

#define EPS (1e-10)

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

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

double distance_ls_p(P a,P b,P c){
	if(dot(b-a,c-a) < EPS) return abs(c-a);
	if(dot(a-b,c-b) < EPS) return abs(c-b);
	return abs(cross(b-a,c-a)) / abs(b-a);
}

bool pol_contains_cir(vector<P> p,P o,double r){
	int n = p.size();
	int cr = 0, cl = 0;
	for(int i=0;i<n;i++){
		P a = p[i];
		P b = p[(i+1)%n];
		if(0 > cross(b-a,o-a)) cr++;
		if(0 < cross(b-a,o-a)) cl++;
		if(distance_ls_p(a,b,o) < r + EPS) return false;
	}
	if(cr != n && cl != n) return false;

	return true;
}

bool cir_contains_pol(vector<P> p,P o,double r){
	for(int i=0;i<p.size();i++){
		if(r < abs(p[i]-o) + EPS) return false;
	}
	return true;
}

bool is_intersected_ls_cir(P a,P b,P o,double r){
	double ao = abs(a-o);
	double bo = abs(b-o);

	return ((ao < r + EPS && r < bo + EPS || r < ao + EPS && bo < r + EPS) ||
					(r < ao + EPS && r < bo + EPS && distance_ls_p(a,b,o) < r + EPS));
}

bool is_intersected_pol_cir(vector<P> p,P o,double r){
	int n = p.size();

	for(int i=0;i<n;i++){
		if(is_intersected_ls_cir(p[i],p[(i+1)%n],o,r)) return true;
	}

	return false;
}

int main(void){
	/*
	P a = P(0,0);
	P b = P(0,2);
	P c = P(-0.1,1);

	cout << cross(b-a,c-a) << endl;
	*/

	while(1){
		vector<P> p;
		for(int i=0;i<3;i++){
			P tmp;
			cin>>tmp.real()>>tmp.imag();
			if(tmp.real()==0 && tmp.imag()==0) return 0;
			p.push_back(tmp);
		}

		P o;
		double r;

		cin>>o.real()>>o.imag()>>r;

		if(pol_contains_cir(p,o,r))
			cout<<"a\n";
		else if(cir_contains_pol(p,o,r)){
			cout<<"b\n";
		}
		else if(is_intersected_pol_cir(p,o,r)){
			cout<<"c\n";
		}
		else{
			cout<<"d\n";
		}
	}
}