#include<iostream>
#include<algorithm>
#include<cstdio>
#include<cmath>
#include<cctype>
#include<math.h>
#include<string>
#include<string.h>
#include<stack>
#include<queue>
#include<vector>
#include<utility>
#include<set>
#include<map>
#include<stdlib.h>
#include<iomanip>
#include<complex>
using namespace std;

#define ll long long
#define ld long double
#define INF 1e9
#define LINF (ll)INF*INF
#define MOD 1000000007
#define rep(i,n) for(int i=0;i<(n);i++)
#define loop(i,a,n) for(int i=a;i<(n);i++)
#define all(in) in.begin(),in.end()
#define shosu(x) fixed<<setprecision(x)

#define int ll //!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

typedef vector<int> vi;
typedef vector<string> vs;
typedef pair<int,int> pii;
typedef vector<pii> vp;

int gcd(int a, int b){
    if(b==0) return a;
    return gcd(b,a%b);
}
int lcm(int a, int b){
    return a/gcd(a,b)*b;
}


const double PI=acos(-1);
const double EPS=1e-9; //うまくいかなかったらゆるめる
typedef complex<double> P;
typedef vector<P> G;
#define shosu(x) fixed<<setprecision(x)
#define diff(P,i) (P[(i+1)%P.size()] - P[i])

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

namespace std{//演算子の定義
    bool operator < (const P& a,const P& b) {
        return real(a)!= real(b) ? real(a) < real(b) : imag(a) < imag(b);
        //return imag(a) != imag(b) ? imag(a) < imag(b) : real(a) < real(b)
    }
    bool operator == (const P& a,const P& b) {
        return a.real()==b.real() && a.imag()==b.imag();
    }
}


void printP(const P &p){
    cout << shosu(10) << p.real() << " " << p.imag() << endl;
}

void printL(const L &l) {
  	cout << shosu(10) << l[0].real() << " " << l[0].imag() << " " << l[1].real() << " " << l[1].imag() << endl;
}

void printG(const G &g) {
    for(int i = 0; i < g.size(); i++){
        cout << shosu(10) << g[i].real() << " " << g[i].imag() << endl;
    }
}



double dot(P a,P b) {
    return real(conj(a)*b);
}

double cross(P a,P b) {
    return imag(conj(a)*b);
}

P orth(const P &a, const P &b) { //点aを(点bを基準として)90度回転した点
    P p = a-b;
    P q(-p.imag(), p.real());
    return q + b;
}

int ccw(P a, P b, P c) { //3点の関係性
    b -= a; c -= a;
    if (cross(b, c) > 0)   return +1;       // counter clockwise
    if (cross(b, c) < 0)   return -1;       // clockwise
    if (dot(b, c) < 0)     return +2;       // c--a--b on line
    if (norm(b) < norm(c)) return -2;       // a--b--c on line
    return 0;                               // a--c--b on line
}

P projection(const L &l, const P &p) { //pの直線l上の射影の点
    double t=dot(p-l[0],l[0]-l[1])/norm(l[0]-l[1]);
    return l[0]+t*(l[0]-l[1]);
}

P reflection(const L &l, const P &p) { //点pの直線lに関して対称な点
    return p+2.0*(projection(l,p)-p);
}

bool isorthogonal(const L &l, const L &m) { //2直線の直交判定
    return fabs(dot(l[1]-l[0], m[1]-m[0])) < EPS;
}

bool isparallel(const L &l, const L &m) { //2直線の平行判定
    return fabs(cross(l[1]-l[0],m[1]-m[0])) < EPS;
}

bool intersectLL(const L &l, const L &m) { //2直線の交差判定
    return !isparallel(l,m);
}

bool intersectSS(const L &s, const L &t) { //2線分の交差判定(完全に交差してないとだめ)
    return ccw(s[0],s[1],t[0])*ccw(s[0],s[1],t[1]) <= 0 &&
         ccw(t[0],t[1],s[0])*ccw(t[0],t[1],s[1]) <= 0;
}

bool intersectSP(const L &s, const P &p) { //直線と点の交差判定
    return abs(s[0]-p)+abs(s[1]-p)-abs(s[1]-s[0]) < EPS; // 三角不等式
}

bool intersectSL(const L &s, const L &l){ //直線と線分の交差判定
    return cross(l[1] - l[0], s[0] - l[0]) * cross(l[1] - l[0], s[1] - l[0]) < EPS;
}


double distancePP(const P &p, const P &q){ //2点間の距離
  return hypot(p.real()-q.real(), p.imag()-q.imag());
}

double distanceSP(const L &s, const P &p) {//線分と点との距離
  const P r = projection(s, p);
  if (intersectSP(s, r)) return abs(r - p);
  return min(abs(s[0] - p), abs(s[1] - p));
}

double distanceSS(const L &s, const L &t) {//2線分の距離
  if (intersectSS(s, t)) return 0;
  return min(min(distanceSP(s, t[0]), distanceSP(s, t[1])),
             min(distanceSP(t, s[0]), distanceSP(t, s[1])));
}

double distanceGL(const G &g, const L &l){//凸多角形と直線の距離
    double ret = LINF;
    int n = g.size();
    rep(i,n){
        L l2(g[i],g[(i+1)%n]);
        ret = min(ret, distanceSS(l2,l));
    }
    return ret;
}

bool intersectCS(const C &c, const L &l){ //円と線分の交差判定
    double close = distanceSP(l,c.c);
    double far = max(distancePP(l[0],c.c),distancePP(l[1],c.c));
    if(close <= c.r && far >= c.r)return true; //重なっていても良い
    //完全に交差している場合のみだと、イコールを外す
    return false;
}

P crosspointSS(const L &a, const L &b) { //2線分の交点
    double t1=abs(cross(a[1]-a[0],b[0]-a[0]));
    double t2=abs(cross(a[1]-a[0],b[1]-a[0]));
    return b[0]+(b[1]-b[0])*t1/(t1+t2);
}

P crosspointLL(const L &l, const L &m) { //2直線の交点
    double A = cross(l[1] - l[0], m[1] - m[0]);
    double B = cross(l[1] - l[0], l[1] - m[0]);
    if (abs(A) < EPS && abs(B) < EPS) return m[0]; // same line
    return m[0] + B / A * (m[1] - m[0]);
}

L crosspointCL(C c,L l){ //円と直線の交点
	P pr = projection(l,c.c);
	P e = (l[1] - l[0]) / abs(l[1] - l[0]);
	double tmp = c.r * c.r - norm(pr - c.c);
	if(abs(tmp) < EPS) tmp = 0;
	double t = sqrt(tmp);
	P a = pr + t * e;
	P b = pr - t * e;
	if(b < a) swap(a,b);
	return L(a,b);
}

L crosspointCC(C a,C b){ //2円の交点
	P tmp = b.c - a.c;
	double d = abs(tmp);
	double q = acos((a.r * a.r + d * d - b.r * b.r)/(2 * a.r * d));
	double t = arg(tmp);//atan(tmp.imag()/tmp.real());
	P p1 = a.c + polar(a.r, t + q);
	P p2 = a.c + polar(a.r, t - q);
	if(p2 < p1) swap(p1,p2);
	return L(p1, p2);
}





P turn(P p,double t){ // 回転
    return p*exp(P(.0,t*PI/180.0));
}


vector<L> tangentCC(C a,C b){ //2円の接線
	if(a.r < b.r) swap(a,b);
	double d = abs(a.c - b.c);
	vector<L> l;
	if(d < EPS) return l;
	if(a.r + b.r < d - EPS){ //離れている
		double t = acos((a.r + b.r)/d);
		t = t * 180 / PI;
		l.push_back(L(a.c + turn(a.r/d*(b.c-a.c), t), b.c + turn(b.r / d * (a.c-b.c), t)));
		l.push_back(L(a.c + turn(a.r/d*(b.c-a.c), -t), b.c + turn(b.r / d * (a.c-b.c), -t)));
	}else if(a.r + b.r < d + EPS){ //外接
		P p = a.c + a.r / d * (b.c - a.c);
		l.push_back(L(p, p + turn(b.c - a.c, 90)));
	}
	if(abs(a.r - b.r) < d - EPS){ //交わっている
		double t1 = acos((a.r - b.r) / d);
		t1 = t1 * 180 / PI;
		double t2 = 180 - t1;
		l.push_back(L(a.c + turn(a.r / d * (b.c-a.c), t1), b.c + turn(b.r / d * (a.c - b.c), -t2)));
		l.push_back(L(a.c + turn(a.r / d * (b.c - a.c), -t1),b.c + turn(b.r / d * (a.c - b.c), t2)));
	}else if(abs(a.r - b.r) < d + EPS){ //内接
		P p = a.c + a.r / d * (b.c - a.c);
		l.push_back(L(p, p + turn(b.c - a.c, 90)));
	}
	return l;
}

L tangentCP(const C &c, const P &p) { //円cの外部にある点pを通るcの接線の接点
    double d = distancePP(c.c,p);
    double r = sqrt(d*d - c.r * c.r);
    C cc(p,r);
    return crosspointCC(c,cc);
}


double area(const G &g) { //多角形の面積
    double S = 0;
    for(int i = 0; i < g.size(); i++){
        S += (cross(g[i],g[(i+1)%g.size()]));
    }
    return abs(S/2.0);
}

bool isconvex(const G &g) { //凸かどうか(全ての内角の大きさが180度以下)
	int n = g.size();
    for(int i = 0; i < n; i++){
        if(ccw(g[(i+n-1)%n], g[i%n], g[(i+1)%n])==-1) return false;
    }
	return true;
}

int inconvex(const G& g, const P& p) { //多角形と点の関係
	bool in = false;
	int n = g.size();
	for(int i = 0; i < n; i++){
		P a = g[i%n] - p;
		P b = g[(i+1)%n] - p;
		if(imag(a) > imag(b)) swap(a, b);
		if(imag(a) < EPS && 0 < imag(b))if(cross(a,b) < 0)in = !in;
		if(abs(cross(a,b)) < EPS && dot(a,b) < EPS) return 1;//ON
	}
	return in ? 2 : 0;//IN : OUT;
}

G convex_hull(G &ps) { // 凸包(点集合 P の全ての点を含む最小の凸多角形)
    int n = ps.size(), k = 0;
	sort(ps.begin(), ps.end());
	G ch(2*n);
	for(int i = 0; i < n; ch[k++] = ps[i++])//lower-hull
		while(k >= 2 && ccw(ch[k-2],ch[k-1],ps[i]) == -1) --k;//<=0  ->  ==-1
	for(int i = n-2,t = k+1; i >= 0; ch[k++] = ps[i--])//upper-hull
		while(k >= t && ccw(ch[k-2],ch[k-1],ps[i]) == -1) --k;//
	ch.resize(k-1);
	return ch;
}

double convex_diameter(const G &pt) { //凸多角形の直径(最遠頂点対間距離)
    const int n = pt.size();
    int is = 0, js = 0;
    for (int i = 1; i < n; ++i) {
        if (imag(pt[i]) > imag(pt[is])) is = i;
        if (imag(pt[i]) < imag(pt[js])) js = i;
    }
    double maxd = norm(pt[is] - pt[js]);

    int i, maxi, j, maxj;
    i = maxi = is;
    j = maxj = js;
    do {
        if (cross(diff(pt,i), diff(pt,j)) >= 0) j = (j+1) % n;
        else i = (i+1) % n;
        if (norm(pt[i] - pt[j]) > maxd) {
            maxd = norm(pt[i] - pt[j]);
            maxi = i; maxj = j;
        }
    } while (i != is || j != js);
    return sqrt(maxd); /* farthest pair is (maxi, maxj). */
}//convex_diameter(g)


G convex_cut(const G& g, const L& l) { //凸多角形の切断
    G Q;
    for (int i = 0; i < g.size(); ++i) {
        P a = g[i], b = g[(i+1)%g.size()];
        if (ccw(l[0], l[1], a) != -1) Q.push_back(a);
        if (ccw(l[0], l[1], a)*ccw(l[0], l[1], b) < 0)
        Q.push_back(crosspointLL(L(a,b), l));
    }
    return Q;
}


L PerpendicularBisector(const L &l) { //線分の垂直二等分線
    P p = l[0];
    P q((l[0].real() + l[1].real())/2.0, (l[0].imag() + l[1].imag())/2.0);
    L t(q, orth(p,q));
    return t;
}

P CircumscribedCircle(const G &g) { //三角形の外接円の半径
    L l1(g[0],g[1]);
    L l2(g[1],g[2]);
    L m1 = PerpendicularBisector(l1);
    L m2 = PerpendicularBisector(l2);
    return crosspointLL(m1, m2);
}

vector<L> getLine(G g){
    int n = g.size();
    vector<L> ret(n);
    rep(i,n)ret[i] = L(g[i],g[(i+1)%n]);
    return ret;
}



P inP(){
    double x,y;
    //scanf("%lf,%lf", &x, &y);
    cin >> x >> y;
    P p(x,y);
    return p;
}


L inL(){
    P p1 = inP();
    P p2 = inP();
    L l(p1,p2);
    return l;
}

C inC(){
    P p = inP();
    double r;
    cin >> r;
    C c(p,r);
    return c;
}


G inG(int n){
    G g(n);
    for(int i = 0; i < n; i++){
        g[i] = inP();
    }
    return g;
}


signed main(void) {
    while(1){
        G g(3);
        g[0] = inP();
        if(g[0].real() == 0 && g[0].imag() == 0)break;
        g[1] = inP();
        g[2] = inP();
        C c = inC();
        vector<L> l = getLine(g);
        if(distancePP(c.c, g[0]) <= c.r &&
            distancePP(c.c, g[1]) <= c.r &&
            distancePP(c.c, g[2]) <= c.r){
                cout << "b" << endl;
        }else if(inconvex(g, c.c) &&
            distanceSP(l[0],c.c) >= c.r &&
            distanceSP(l[1],c.c) >= c.r &&
            distanceSP(l[2],c.c) >= c.r){
                cout << "a" << endl;
        }else if(intersectCS(c, l[0]) ||
                intersectCS(c, l[1]) ||
                intersectCS(c, l[2])){
                cout << "c" << endl;
        }else{
            cout << "d" << endl;
        }



    }

}

