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 <cstdio>
#include <complex>
#include <vector>
#include <algorithm>
#include <cmath>
using namespace std;

#define EPS 1e-8
typedef complex<double> P;

struct L : public vector<P> {
  L(const P &a, const P &b) {
    push_back(a); push_back(b);
  }
};

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

double cross(const P& a, const P& b) {
  return imag(conj(a)*b);
}

double dot(const P& a, const P& b) {
  return real(conj(a)*b);
}

double dist(const P& a, const P& b)
{
	return sqrt((a.real()-b.real())*(a.real()-b.real())+(a.imag()-b.imag())*(a.imag()-b.imag()));
}

bool isIntoC(const P& p, const C& c)
{
	return dist(p, c.p) < c.r+EPS;
}

bool isIntoC(const L& l, const C& c)
{
	return dist(l[0], c.p) < c.r+EPS && dist(l[1], c.p) < c.r+EPS;
}

bool intersectSC(const L& l, const C& c)
{
	if(isIntoC(l, c)) return false;
	
	double vc=dot(l[1]-l[0], c.p-l[0]);
	if(vc+EPS<0)
	{
		return dist(l[0], c.p)+EPS<c.r;
	}
	
	double vv=dot(l[1]-l[0], l[1]-l[0]);
	
	if(vc > vv)
	{
		double d=dist(l[1], c.p);
		
		return d*d+EPS<c.r*c.r;
	}
	
	double cc=dot(c.p-l[0], c.p-l[0]);
	
	return cc-(vc/vv)*vc+EPS < c.r*c.r;
}

int main()
{
	int x,y,r;
	while(scanf("%d%d", &x, &y), (x||y))
	{
		vector<P> tri;
		for(int i=0; i<3; i++)
		{
			tri.push_back(P(x,y));
			if(i==2) break;
			scanf("%d%d", &x, &y);
		}
	
		scanf("%d%d%d", &x,&y,&r);
		C c(P(x,y), r);
		
		bool g=true;
		for(int i=0; i<3; i++)
		{
			if(!isIntoC(tri[i], c))
			{
				g=false;
				break;
			}
		}
		
		if(g) 
		{
			puts("b");
			continue;
		}
		
		g=false;
		for(int i=0; i<tri.size(); i++)
		for(int j=i+1; j<tri.size(); j++)
		{
			if(intersectSC(L(tri[i], tri[j]), c))
			{
				g=true;
			}
		}
		if(g)
		{
			puts("c");
			continue;
			
		}
		int cnt=0, cr[3];
		cr[0]=cross(tri[0]-tri[1], tri[0]-c.p);
		cr[1]=cross(tri[1]-tri[2], tri[1]-c.p);
		cr[2]=cross(tri[2]-tri[0], tri[2]-c.p);
		
		if(cr[0]>EPS&&cr[1]>EPS&&cr[2]>EPS) puts("a");
		else puts("d");
	}
}

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 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;
	return abs(nm) / sqrt(de);
}

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

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;
	}

	P p0 = tri[0] - circle.first;
	P p1 = tri[1] - circle.first;
	P p2 = tri[2] - circle.first;
	d[3] = norm(p0);
	d[4] = norm(p1);
	d[5] = norm(p2);
	// 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();
	}
}