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 <cstdlib>
#include <cmath>
#include <climits>
#include <cfloat>
#include <map>
#include <utility>
#include <ctime>
#include <set>
#include <iostream>
#include <memory>
#include <string>
#include <cstring>
#include <vector>
#include <algorithm>
#include <functional>
#include <fstream>
#include <sstream>
#include <complex>
#include <stack>
#include <queue>
#include <cstring>
#include <numeric>
#include <cassert>
using namespace std;
static const double EPS = 1e-10;
typedef long long ll;
#define rep(i,n) for(int i=0;i<n;i++)
#define rev(i,n) for(int i=n-1;i>=0;i--)
#define all(a) a.begin(),a.end()
#define mp(a,b) make_pair(a,b)
#define pb(a) push_back(a)
#define SS stringstream
#define DBG1(a) rep(_X,sz(a)){printf("%d ",a[_X]);}puts("");
#define DBG2(a) rep(_X,sz(a)){rep(_Y,sz(a[_X]))printf("%d ",a[_X][_Y]);puts("");}
#define bitcount(b) __builtin_popcount(b)
#define EACH(i,c) for(typeof((c).begin()) i=(c).begin(); i!=(c).end(); ++i)

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

int chk(P a,P b,double r){
	double t = abs(a-b) - r;
	if( fabs(t) < EPS ) return 0;
	return t < 0 ? -1 : 1;
}
double cross(P a,P b){
	return imag(a*conj(b));
}
double dot(const P& a, const P& b) {return real(conj(a)*b);}



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]);}

bool intersectSP(const L &s, const P &p) {return abs(s[0]-p)+abs(s[1]-p)-abs(s[1]-s[0]) < EPS;}

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));
}

int main(){
	P a,b,c,d;
	double r;
	while(cin >> a.real() >> a.imag() , a.real() != 0 || a.imag() != 0 ){
		cin >> b.real() >> b.imag();
		cin >> c.real() >> c.imag();
		cin >> d.real() >> d.imag();
		cin >> r;
		
		int in = 0;
		int mk = cross(a-d,b-d) <= 0 ;
		if( mk == (cross(b-d,c-d) <= 0) && mk == (cross(c-d,a-d) <= 0)){
			in = 1;
		}
		
		if( chk(a,d,r) <= 0 &&  chk(b,d,r) <= 0 &&  chk(c,d,r) <= 0 ){
			cout << "b" << endl;
		}else if( chk(a,d,r) >= 0 &&  chk(b,d,r) >= 0 &&  chk(c,d,r) >= 0){
			if( in ){
				cout << "a" << endl;
			}else if( distanceSP(L(a,b),d) < r - EPS || distanceSP(L(b,c),d) < r - EPS || distanceSP(L(a,c),d) < r - EPS  ){
				cout << "c" << endl;
			}else{
				cout << "d" << endl;
			}
		}else{
			cout << "c" << endl;
		}
	

		
	}
}

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++)

using namespace std;

bool f(double x,double y,double x0,double y0,double x4,double y4,double r){
double p=(x0-x)*(y0-y4)-(x0-x4)*(y0-y);
double q=(y0-y)*(y0-y)+(x0-x)*(x0-x);
if(x-x0!=0 && y-y0!=0){
if(abs(p)/sqrt(q)<=r && (-p*(y0-y)/q)>min(x,x0) && (-p*(y0-y)/q)<max(x,x0))
return true;
else return false;
}
if(x-x0==0 && abs(x0)<=r && sqrt(r*r-x0*x0)<max(y,y0) && -sqrt(r*r-x0*x0)>min(y,y0))
return true;
else return false;
if(y-y0==0 && abs(y0)<=r && sqrt(r*r-y0*y0)<max(x,x0) && -sqrt(r*r-y0*y0)>min(x,x0))
return true;
else return false;
}

bool g(double x1,double y1,double x2,double y2,double x3,double y3,double x4,double y4){
double p=(y1-y4)/(x1-x4);
double q=(y2-y3)/(x2-x3);
double u=((y3-y4)-q*(x3-x4))/(p-q)-x1+x4;
double v=(p*(y3-y4)-p*q*(x3-x4))/(p-q)-y1+y4;
if((x1-x4)!=0 && (x2-x3)!=0 && (p-q)!=0 && (u*u+v*v)>((x1-x4)*(x1-x4)+(y1-y4)*(y1-y4)))
return true;
else return false;
if((x1-x4)==0 && (x2-x3)!=0 && abs(y1-y4)<abs(y1-y3+q*(x3-x4)))
return true;
else return false;
if((x2-x3)==0 && (x1-x4)!=0 && ((x1-x4)*(x1-x4)+(y1-y4)*(y1-y4))<((x2-x1)*(x2-x1)+(p*x2-p*x4-y1+y4)*(p*x2-p*x4-y1+y4)))
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) && g(x2,y2,x1,y1,x3,y3,x4,y4) && g(x3,y3,x2,y2,x1,y1,x4,y4))
cout << "a" << endl;

else cout << "d" << endl;

}


}

	return 0;
}