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>
using namespace std;

//objects-----------------------------------------------------------------
typedef complex<double>Point;
typedef Point Vector;
typedef vector<Point>Polygon;

struct Segment{
    Point p1,p2;
    Segment(const Point &p1=Point(),const Point &p2=Point()):p1(p1),p2(p2){}
};

struct Line{
    Point p1,p2;
    Line(const Point &p1=Point(),const Point &p2=Point()):p1(p1),p2(p2){}
};

struct Circle{
    Point p;
    double r;
    Circle(const Point &p=Point(),double r=0.0):p(p),r(r){}
};

//constants---------------------------------------------------------------
const double PI=acos(-1);
const double EPS=1e-12;
const double INF=1e16;
const int COUNTER_CLOCKWISE=1;
const int CLOCKWISE=-1;
const int ONLINE_BACK=2;
const int ONLINE_FRONT=-2;
const int ON_SEGMENT=0;
const int IN=2;
const int ON=1;
const int OUT=0;
//calculations------------------------------------------------------------
inline double square(double a){return a*a;}
inline double norm(const Point &a){return square(a.real())+square(a.imag());}
inline double dot(const Point &a,const Point &b){return (conj(a)*b).real();}
inline double cross(const Point &a,const Point &b){return (conj(a)*b).imag();}
inline double toDeg(double t){return t/PI*180.0;}
inline double toRad(double t){return t/180.0*t;}
inline double arg(Vector p){return atan2(imag(p),real(p));}
inline Vector polar(double a,double r){return Vector(cos(r),sin(r))*a;}
#define equals(a,b) (fabs((a)-(b))<EPS)
#define equalsv(a,b) (equals(real(a),real(b))&&equals(imag(a),imag(b)))
#define curr(P,i) P[(i)%P.size()]
#define next(P,i) P[(i+1)%P.size()]
#define prev(P,i) P[(i-1+P.size())%P.size()]

int ccw(Point p0,Point p1,Point p2){
    Vector a=p1-p0;
    Vector b=p2-p0;
    if(cross(a,b)>EPS)return COUNTER_CLOCKWISE;
    if(cross(a,b)<-EPS)return CLOCKWISE;
    if(dot(a,b)<EPS)return ONLINE_BACK;
    if(norm(a)<norm(b))return ONLINE_FRONT;
    return ON_SEGMENT;
}

//inputer-----------------------------------------------------------------
void inputPoint(Point &p){
    double x,y;
    scanf("%lf%lf",&x,&y);
    p=Point(x,y);
}

void inputSegment(Segment &s){
    inputPoint(s.p1);
    inputPoint(s.p2);
}

void inputLine(Line &l){
    inputPoint(l.p1);
    inputPoint(l.p2);
}

void inputCircle(Circle &c){
    inputPoint(c.p);
    scanf("%lf",&c.r);
}

void inputPolygon(Polygon &g,int n){
    g.resize(n);
    for(int i=0;i<n;i++)inputPoint(g[i]);
}

//orthogonal--------------------------------------------------------------
bool orthogonal(Vector a,Vector b){
    return equals(dot(a,b),0.0);
}

bool orthogonal(Point a1,Point a2,Point b1,Point b2){
    return orthogonal(a1-a2,b1-b2);
}

bool orthogonal(Segment s1,Segment s2){
    return orthogonal(s1.p1-s1.p2,s2.p1-s2.p2);
}

bool orthogonal(Line l1,Line l2){
    return orthogonal(l1.p1-l1.p2,l2.p1-l2.p2);
}

//parallel----------------------------------------------------------------
bool parallel(Vector a,Vector b){
    return equals(cross(a,b),0.0);
}

bool parallel(Point a1,Point a2,Point b1,Point b2){
    return parallel(a1-a2,b1-b2);
}

bool parallel(Segment s1,Segment s2){
    return parallel(s1.p1-s1.p2,s2.p1-s2.p2);
}

bool parallel(Line l1,Line l2){
    return parallel(l1.p1-l1.p2,l2.p1-l2.p2);
}

//project&reflect---------------------------------------------------------
Point project(Line s,Point p){
    Vector base=s.p2-s.p1;
    double r=dot(p-s.p1,base)/norm(base);
    return s.p1+base*r;
}

Point reflect(Line s,Point p){
    return p+(project(s,p)-p)*2.0;
}

//distance----------------------------------------------------------------
double distancePP(Point a,Point b){
    return abs(a-b);
}

double distanceLP(Line l,Point p){
    return fabs(cross(l.p2-l.p1,p-l.p1)/abs(l.p2-l.p1));
}

double distanceLL(Line l,Line m){
    if(parallel(l,m))return distanceLP(l,m.p1);
    return 0.0;
}

double distanceSP(Segment s,Point p){
    if(dot(s.p2-s.p1,p-s.p1)<0.0)return distancePP(p,s.p1);
    if(dot(s.p1-s.p2,p-s.p2)<0.0)return distancePP(p,s.p2);
    return distanceLP(Line(s.p1,s.p2),p);
}

double distanceSS(Segment s1,Segment s2){
    if(ccw(s1.p1,s1.p2,s2.p1)*ccw(s1.p1,s1.p2,s2.p2)<=0&&
        ccw(s2.p1,s2.p2,s1.p1)*ccw(s2.p1,s2.p2,s1.p2)<=0)return 0;
    return min(min(distanceSP(s1,s2.p1),distanceSP(s1,s2.p2)),
            min(distanceSP(s2,s1.p1),distanceSP(s2,s1.p2)));
}

double distanceCS(Circle c,Segment s){
    return distanceSP(s,c.p)-c.r;
}

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

bool intersectSS(Segment s1,Segment s2){
    return intersectSS(s1.p1,s1.p2,s2.p1,s2.p2);
}

bool intersectCL(Circle c,Line l){
    return fabs(cross(l.p2-l.p1,c.p-l.p1)/abs(l.p2-l.p1))<=c.r+EPS;
}

bool intersectCC(Circle c1,Circle c2){
    return abs(c1.p-c2.p)<=c1.r+c2.r+EPS;
}

bool intersectCS(Circle c,Segment s){
    return distanceSP(s,c.p)<=c.r+EPS;
}

bool intersectGC(Polygon g,Circle c){
    for(int i=0;i<g.size();i++){
        Segment s(g[i],next(g,i));
        if(intersectCS(c,s))return true;
    }
    return false;
}

//crossPoint--------------------------------------------------------------
Point crossPoint(Segment s1,Segment s2){
    double crs=cross(s1.p2-s1.p1,s2.p2-s2.p1);
    return s2.p1+(s2.p2-s2.p1)*cross(s1.p2-s1.p1,s1.p2-s2.p1)*(1.0/crs);
}

pair<Point,Point>crossPoints(Circle c,Line l){
    assert(intersectCL(c,l));
    Vector pr=project(l,c.p);
    Vector e=(l.p2-l.p1)/abs(l.p2-l.p1);
    double base=sqrt(square(c.r)-norm(pr-c.p));
    return make_pair(pr+e*base,pr-e*base);
}

pair<Point,Point>crossPoints(Circle c1,Circle c2){
    assert(intersectCC(c1,c2));
    double d=abs(c1.p-c2.p);
    double a=acos((square(c1.r)+square(d)-square(c2.r))/(2*c1.r*d));
    double t=arg(c2.p-c1.p);
    return make_pair(c1.p+polar(c1.r,t+a),c1.p+polar(c1.r,t-a));
}

//polygon------------------------------------------------------------------
double area(Polygon &g){
    const int sz=g.size();
    double res=cross(g[sz-1],g[0]);
    for(int i=1;i<sz;i++)res+=cross(g[i-1],g[i]);

    return fabs(res)*0.5;
}

bool convex(Polygon &g){
    for(int i=0;i<g.size();i++){
        if(ccw(prev(g,i),curr(g,i),next(g,i))==CLOCKWISE)return false;
    }
    return true;
}

//other-------------------------------------------------------------------
int contains(Circle c,Point p){
    double d=distancePP(c.p,p);
    if(equals(d,c.r))return ON;
    return d<c.r?IN:OUT;
}

int contains(Polygon g,Point p){
    int n=g.size();
    bool x=false;
    for(int i=0;i<n;i++){
        Point a=g[i]-p,b=next(g,i)-p;
        if(abs(cross(a,b))<EPS&&dot(a,b)<EPS)return ON;
        if(imag(a)>imag(b))swap(a,b);
        if(imag(a)<EPS&&EPS<imag(b)&&cross(a,b)>EPS)x=!x;
    }
    return x?IN:OUT;
}

int contains(Circle c,Polygon g){
    for(int i=0;i<g.size();i++){
        if(!contains(c,g[i]))return OUT;
    }
    return IN;
}

int contains(Polygon g,Circle c){
    if(contains(g,c.p)!=IN)return OUT;
    for(int i=0;i<g.size();i++){
        if(distanceCS(c,Segment(g[i],next(g,i)))<-EPS)return OUT;
    }
    return IN;
}
//end---------------------------------------------------------------------

int main(){
    while(true){
        Polygon g(3);
        Circle c;
        inputPoint(g[0]);
        if(real(g[0])==0&&imag(g[0])==0)break;
        inputPoint(g[1]);
        inputPoint(g[2]);

        inputCircle(c);

        if(contains(g,c))puts("a");
        else if(contains(c,g))puts("b");
        else if(intersectGC(g,c))puts("c");
        else puts("d");
    }
    return 0;
}