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 "bits/stdc++.h"
#include <cassert>
using namespace std;
typedef long long ll;
typedef pair<int,int> pii;
#define rep(i,n) for(ll i=0;i<(ll)(n);i++)
#define all(a)  (a).begin(),(a).end()
#define vi vector<int>
#define pb push_back
#define INF 999999999
//#define INF (1LL<<59)

#define OUT 0
#define ON 1
#define IN 2
#define EPS (1e-10)
class P{                    //???
public:
    double x,y;
    
    P(double _x=0,double _y=0):x(_x),y(_y){};
    P operator + (P &p     ){ return P( x+p.x , y+p.y ); }  //??????
    P operator - (P &p     ){ return P( x-p.x , y-p.y ); }  //??????
    P operator * (double k ){ return P( x*k   , y*k    ); } //??????
    P operator / (double k ){ return P( x/k   , y/k    ); } //??????
    
    bool operator == (const P &p){ return ( fabs(x-p.x)<EPS && fabs(y-p.y)<EPS ); }
    //    bool operator < (const P &p)const{ return ( x!=p.x ? x<p.x:y<p.y ); }
    
    double norm(){ return x*x+y*y; }        //?????????
    double abs() { return sqrt(norm()); }   //??§??????
    
};
struct C{P p;double r;};    //???
struct L{P p1,p2;};         //??´???
typedef vector<P> Polygon;  //????§???¢
typedef P Vector;           //????????????

double dot  (Vector a,Vector b) { return a.x*b.x+a.y*b.y; }
double cross(Vector a,Vector b) { return a.x*b.y-a.y*b.x; }
double sqDist(P a, P b)         {return (a.x-b.x)*(a.x-b.x)+(a.y-b.y)*(a.y-b.y);}
double dist (P a, P b)          {return sqrt(sqDist(a,b));}
Vector vec(L a)                 {return P(a.p2.x-a.p1.x,a.p2.y-a.p1.y);}

int ccw(P p0,P p1,P p2){    //AOJ_BOOK_P386 verified
    Vector a = p1 - p0;
    Vector b = p2 - p0;
    
    if( cross(a,b) > EPS  ) return 1 ;      //COUNTER_CLOCKWISE
    if( cross(a,b) < -EPS ) return -1;      //CLOCKWISE
    if( dot(a,b)   < -EPS ) return 2;       //ONLINE_BACK
    if( a.norm()   < b.norm() ) return -2;  //ONLINE_FRONT
    
    return 0;                               //ON_SEGMENT;
}

//??´?????´????????? verified AOJ0058
bool orthogonal(P p1,P p2,P p3,P p4){ return abs(dot(p1-p2,p3-p4))<EPS; }

//??????????????????
bool intersect(P p1,P p2,P p3,P p4){ return ( ccw(p1,p2,p3)*ccw(p1,p2,p4)<=0 && ccw(p3,p4,p1)*ccw(p3,p4,p2)<=0 ); }

//????????¨???????????¢ verified ARC042-B
double dLP(L l, P p) { return abs(cross(l.p2-l.p1, p-l.p1)) /(l.p2-l.p1).abs(); }

//??´?????¨??´????????¢??? verified AOJ CGL_2
int rLL(L a,L b){
    if( cross(vec(a),vec(b))==0 ) return 2; //??????
    if( dot(vec(a),vec(b))==0 )   return 1; //?????´
    return 0;
}

// ?????¨??????????????????
int contains(C c, P p){
    double d = (c.p-p).abs();
    if(d-c.r>EPS) return OUT;
    if(abs(d-c.r)<EPS) return ON;
    return IN;
}

//??´?????¨??????????????????(????????£???????????´??????????????±???)
bool intersect_circle_(P center, double r, L line){
    if( dLP(line,center) <= r+EPS ) return true;
    return false;
}

//????????¨???????????¢ verified QUPC-G
double dSP(L s, P p){
    if(dot((s.p2-s.p1)   , p-s.p1) <= EPS) return (p - s.p1).abs();
    if(dot((s.p2-s.p1)*-1, p-s.p2) <= EPS) return (p - s.p2).abs();
    return dLP(s, p);
}

//????????¨??????????????????(????????£???????????´??????????????±???) verified QUPC-G
bool iCS(C c, L l){
    int c1 = contains(c, l.p1);
    int c2 = contains(c, l.p2);
    if(c1 > c2) swap(c1, c2);
    
    // (OUT, OUT) (OUT, ON) (OUT, IN) (ON, ON) (ON, IN) (IN, IN) ???6??????
    if(c1 == OUT && c2 == IN) return true;
    if(c1 == IN  && c2 == IN) return false;
    if(c1 == ON) return true; // (??\????????¨???)
    double d = dSP(l, c.p);
    if(d-c.r<-EPS) return true;
    if(d-c.r>EPS) return false;
    return true; // (??\????????¨???)
}

//????´?????§???¢??????
bool isSimple( Polygon pol ){
    //???????????????????????????pol?????\????????????
    size_t pol_size = pol.size()-1;
    rep(i,pol_size){
        for(int j=i+2;j<pol_size;j++){
            if( i==j || i==(j-1+pol_size)%pol_size || i==(j+1+pol_size)%pol_size )continue;
            if( intersect(pol[i], pol[i+1], pol[j], pol[j+1]) )return false;
        }
    }
    return true;
}

//?????????????§???¢????????´?????????????????????????±??????? true???????????´ verified AOJ0012
int isPointInsidePolygon(vector<P> pol,P p){
    int c=0;
    rep(i,pol.size()){
        if( cross(pol[i]-pol[(i+1)%pol.size()] , p-pol[(i+1)%pol.size()])==0 ) return ON;
        if( cross(pol[i]-pol[(i+1)%pol.size()] , p-pol[(i+1)%pol.size()])>0 ) c++;
    }
    if( c%pol.size() )return OUT;
    return IN;
}

//?????¨???????§???¢???????????¶???????????????
int CPOLarea(C c ,Polygon pol){
    vector<L> lines;
    vector<int> res(pol.size());
    bool POLinC=true, isFar=true;
    
    rep(i,pol.size()){
        if( contains(c,pol[i])==OUT )POLinC=false;
        res[i] = contains(c,pol[i]);
        lines.pb( L{ pol[i] , pol[(i+1)%pol.size()] });
        if( sqDist(c.p,pol[i])<c.r*c.r )isFar=false;
    }
    
    if( POLinC )                                        return 2;   //????§???¢??????????????¨b
    if( isPointInsidePolygon(pol, c.p)==IN && isFar )   return 3;   //????§???¢????????¨??????a
    rep(i,lines.size()) if( iCS(c,lines[i]) )           return 1;   //????§???¢??¨????????????c
    return 0;
}

//?????? verified AOJ0068,QUPC-G
//???????????§??????
bool cmp_x(const P& p, const P& q){
    if(p.x != q.x)return p.x<q.x;
    return p.y<q.y;
}

//??????????±???????
vector<P> convex_hull(vector<P> ps){
    int n = ps.size();
    sort(all(ps),cmp_x);
    int k=0;            //?????????????????°
    vector<P> qs(n*2);  //?§????????????????
    //?????´???????????????
    rep(i,n){
        while( k>1 && cross((qs[k-1]-qs[k-2]) , (ps[i]-qs[k-1]))<=0 ) k--;
        qs[k++]=ps[i];
    }
    //?????´???????????????
    for(int i=n-2, t=k;i>=0;i--){
        while( k>t && cross((qs[k-1]-qs[k-2]) , (ps[i]-qs[k-1]))<=0 ) k--;
        qs[k++]=ps[i];
    }
    qs.resize(k-1);
    return qs;
}

int main(){
    int a,b;
    while(cin>>a>>b&&(a||b)){
        Polygon pol(3);
        pol[0] = P(a,b);
        rep(i,2)cin>>pol[i+1].x>>pol[i+1].y;
        C c;
        cin>>c.p.x>>c.p.y>>c.r;
        int res =  CPOLarea(c,pol);
        
        cout<<(char)('a'+(3-res))<<endl;
    }
}