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++)

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

else cout << "d" << endl;

}


}

	return 0;
}

C++: #include<stdio.h>
#include<algorithm>
using namespace std;
bool CL_intersect(int ax,int ay,int bx,int by,int cx,int cy,int r){
	int acx=cx-ax;
	int acy=cy-ay;
	int bcx=cx-bx;
	int bcy=cy-by;
	int abx=bx-ax;
	int aby=by-ay;
	if(acx*acx+acy*acy>=r*r||bcx*bcx+bcy*bcy>=r*r){
		if(abx*acx+aby*acy<0&&acx*acx+acy*acy<=r*r)return true;
		else if(abx*acx+aby*acy>abx*abx+aby*aby&&bcx*bcx+bcy*bcy<=r*r)return true;
		else if(acx*acx+acy*acy-(long long)(abx*acx+aby*acy)*(abx*acx+aby*acy)/(abx*abx+aby*aby)<=r*r)return true;
	}
	return false;
}
bool intersect(int ax,int ay,int bx,int by,int cx,int cy,int dx,int dy){
	long long ta=(cx-dx)*(ay-cy)+(cy-dy)*(cx-ax);
	long long tb=(cx-dx)*(by-cy)+(cy-dy)*(cx-bx);
	long long tc=(ax-bx)*(cy-ay)+(ay-by)*(ax-cx);
	long long td=(ax-bx)*(dy-ay)+(ay-by)*(ax-dx);
	if(ta*tb<=0LL&&tc*td<=0LL)return true;
	else return false;
}
int main(){
	int a,b,c,d,e,f,g,h,r;
	while(scanf("%d%d",&a,&b),a||b){
		scanf("%d%d%d%d%d%d%d",&c,&d,&e,&f,&g,&h,&r);
		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)printf("b\n");
		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))printf("c\n");
		else if(intersect(g,h,a,b,c,d,e,f)||intersect(g,h,c,d,a,b,e,f)||intersect(g,h,e,f,a,b,c,d))printf("d\n");
		else printf("a\n");
	}
}

C++: #include<stdio.h>
#include<algorithm>
using namespace std;
bool CL_intersect(int ax,int ay,int bx,int by,int cx,int cy,int r){
	int acx=cx-ax;
	int acy=cy-ay;
	int bcx=cx-bx;
	int bcy=cy-by;
	int abx=bx-ax;
	int 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-(long long)(abx*acx+aby*acy)*(abx*acx+aby*acy)/(abx*abx+aby*aby)<=r*r)return true;
		}
	}
	return false;
}
bool inclusive(int ax,int ay,int bx,int by,int cx,int cy,int px,int py){
	bool ok1=true;
	bool 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;
}
int main(){
	int a,b,c,d,e,f,g,h,r;
	while(scanf("%d%d",&a,&b),a||b){
		scanf("%d%d%d%d%d%d%d",&c,&d,&e,&f,&g,&h,&r);
		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)printf("b\n");
		else if(inclusive(a,b,c,d,e,f,g,h))printf("a\n");
		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))printf("c\n");
		else printf("d\n");
	}
}