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 <iostream>
#include <complex>

using namespace std;

#define dump(n) cerr<<"# "<<#n<<"="<<(n)<<endl

const double EPS=1e-9;
const double INFTY=1e12;
typedef complex<double> Point;

struct Line{
	Point pos,dir;
	Line(){}
	Line(Point p,Point d):pos(p),dir(d){}
};

int Signum(double x)
{
	return abs(x)<EPS?0:x>0?1:-1;
}

double Dot(Point a,Point b)
{
	return real(conj(a)*b);
}

double Cross(Point a,Point b)
{
	return imag(conj(a)*b);
}

int CCW(Point a,Point b,Point c)
{
	Point d1=b-a,d2=c-a;
	int sign=Signum(Cross(d1,d2));
	
	if(sign)
		return sign;	// 1:ccw,-1:cw
	if(Dot(d1,d2)<-EPS)
		return -2;		// c-a-b
	if(norm(d1)<norm(d2)-EPS)
		return 2;		// a-b-c
	return 0;			// a-c-b
}

// ツ点ツづ陳シツ静シツづ個仰猟猟」
double DistancePL(Point p,Line l)
{
	return abs(Cross(p-l.pos,l.dir))/abs(l.dir);
}
// ツ点ツづ静シツ閉ェツづ個仰猟猟」
double DistancePS(Point p,Line s)
{
	Point a=p-s.pos,b=s.dir;
	Point c=b*Dot(a,b)/norm(b);
	int pos=CCW(Point(0,0),b,c);
	if(pos==-2 || pos==2)
		return min(abs(a),abs(b-a));
	return abs(a-c);
}

void solve(Point p[],Point c,double r)
{
	{	// triangle in circle
		double dist=0;
		for(int i=0;i<3;i++)
			dist=max(dist,norm(p[i]-c));
		if(dist<=r*r){
			puts("b");
			return;
		}
	}
	
	{	// circle in triangle
		bool res=true;
		int ccw=0;
		for(int i=0;i<3;i++){
			int temp=CCW(p[i],p[(i+1)%3],c);
			if(temp==-2 || temp==2)
				continue;
			res&=temp*ccw>=0;
			ccw=temp;
		}
		if(res){
			double dist=INFTY;
			for(int i=0;i<3;i++)
				dist=min(dist,DistancePL(c,Line(p[i],p[(i+1)%3]-p[i])));
			if(Signum(dist-r)>=0){
				puts("a");
				return;
			}
		}
	}
	
	{	// intersect
		double dist=INFTY;
		for(int i=0;i<3;i++)
			dist=min(dist,DistancePS(c,Line(p[i],p[(i+1)%3]-p[i])));
		if(Signum(dist-r)<0){
			puts("c");
			return;
		}
	}
	
	puts("d");
}

int main()
{
	for(;;){
		Point p[3],c;
		double r;
		for(int i=0;i<3;i++)
			cin>>real(p[i])>>imag(p[i]);
		cin>>real(c)>>imag(c)>>r;
		if(norm(p[0])==0)
			break;
		
		//for(int i=0;i<3;i++)
		//	printf("(%.3f,%.3f) ",real(p[i]),imag(p[i]));
		//printf("(%.3f,%.3f,%.3f)\n",real(c),imag(c),r);
		
		solve(p,c,r);
	}
	
	return 0;
}

C++: #include <cstdio>
#include <cstdlib>
#include <cmath>
#include <climits>
#include <cfloat>
#include <map>
#include <utility>
#include <set>
#include <iostream>
#include <memory>
#include <string>
#include <vector>
#include <algorithm>
#include <functional>
#include <sstream>
#include <complex>
#include <stack>
#include <queue>
typedef long long ll;
using namespace std;
 
struct Pos{
    double x,y;
    Pos(double x,double y) : x(x) , y(y) {}
    Pos(){}
};
 
struct Triangle{
    Pos pos[3];
};
struct Circle{
    Pos pos;
    double r;
};
 
static const double kyuriEPS = 1e-4;
typedef std::complex<double> myPoint;
double dot(const myPoint& a, const myPoint& b) {return real(conj(a)*b);}
struct myL : public std::vector<myPoint> {
    myL(const myPoint &a, const myPoint &b) {push_back(a); push_back(b);}
};
 
bool intersectSP(const myL &s, const myPoint &p) {
    return abs(s[0]-p)+abs(s[1]-p)-abs(s[1]-s[0]) < kyuriEPS;
} 
myPoint projection(const myL &l, const myPoint &p) {
    double t = dot(p-l[0], l[0]-l[1]) / norm(l[0]-l[1]);return l[0] + t*(l[0]-l[1]);
}
 
double distanceSP(const myL &s, const myPoint &p) {
    const myPoint r = projection(s, p);
    if (intersectSP(s, r)) return abs(r - p);
    return std::min(abs(s[0] - p), abs(s[1] - p));
}
//ct
int judgeHit(const Circle &c,const Triangle &t){
    int counter = 0 , ctt = 0;
    for(int i = 0 ; i < 3 ; i++){
        if( abs(myPoint(c.pos.x,c.pos.y)-myPoint(t.pos[i].x,t.pos[i].y)) < c.r - kyuriEPS ){
            ctt++;
        }
    }
    if( ctt == 3 ) return 1;
    for(int i = 0 ; i < 3 ; i++){
        double dist = distanceSP(myL(myPoint(t.pos[i].x,t.pos[i].y),myPoint(t.pos[(i+1)%3].x,t.pos[(i+1)%3].y)),myPoint(c.pos.x,c.pos.y));
        if( dist <= c.r + kyuriEPS){
            counter++;
        }
    }
    if( counter >= 1 ) return 2;
     
    double tS2 = fabs( (t.pos[1].x-t.pos[0].x)*(t.pos[2].y-t.pos[0].y) - (t.pos[1].y-t.pos[0].y)*(t.pos[2].x-t.pos[0].x) );
     
    // a.pos[i] on B
    double area = 0;
    for(int j = 0 ; j < 3 ; j++){
        area += fabs((t.pos[j].x-c.pos.x)*(t.pos[(j+1)%3].y-c.pos.y) - (t.pos[j].y-c.pos.y)*(t.pos[(j+1)%3].x-c.pos.x));
    }
    if( fabs( fabs(area) - tS2 ) < kyuriEPS ) return 0;  
    else return 3;
}
 
int main(){
    Triangle t;
    Circle c;
    while(cin >> t.pos[0].x >> t.pos[0].y && (t.pos[0].x||t.pos[0].y)){
        for(int i = 1 ; i < 3 ; i++)
            cin >> t.pos[i].x >> t.pos[i].y;
        cin >> c.pos.x >> c.pos.y >> c.r;
        cout << (char)('a'+judgeHit(c,t)) << endl;
    }
}