#include <bits/stdc++.h>
using namespace std;

const double EPS = 1e-6;

typedef complex<double> C;

double ip(C a, C b){
    return a.real() * b.real() + a.imag() * a.imag();
}

double op(C a, C b){
    return a.real() * b.imag() - a.imag() * b.real();
}

double dist_lc(C p, double a, double b, double c){
    return abs(a * p.real() + b * p.imag() + c) / abs(p);
}

int main(){
    C p1, p2, p3, pr;
    double r;
    bool flg1, flg2, flg3, flg4;
    while(true){
        int s, t;
        double x1, y1, x2, y2, x3, y3, xr, yr;
        scanf("%d%d", &s, &t);
        if(s == 0 && t == 0) break;
        p1.real(s); p1.imag(t);
        x1 = p1.real(); y1 = p1.imag();
        scanf("%d%d", &s, &t);
        p2.real(s); p2.imag(t);
        x2 = p2.real(); y2 = p2.imag();
        scanf("%d%d", &s, &t);
        p3.real(s); p3.imag(t);
        x3 = p3.real(); y3 = p3.imag();
        scanf("%d%d", &s, &t);
        pr.real(s); pr.imag(t);
        xr = s; yr = t;
        scanf(" %lf", &r);
        if(abs(p1 - pr) <= r && abs(p2 - pr) <= r && abs(p3 - pr) <= r){
            puts("b");
            continue;
        }
        double a = x1 - x2, b = y1 - y2;
        double c = x2 - x3, d = y2 - y3;
        double e = x3 - x1, f = y3 - y1;
        flg1 = abs(b * xr - a * yr - b * x1 + a * y1)/abs(p1-p2) >= r;
        flg2 = abs(d * xr - c * yr - d * x2 + c * y2)/abs(p2-p3) >= r;
        flg3 = abs(f * xr - e * yr - f * x3 + e * y3)/abs(p3-p1) >= r;
        flg4 = op(p1-pr, p2-pr) + op(p2-pr, p3-pr) + op(p3-pr, p1-pr) > EPS;
        if(flg1 && flg2 && flg3 && flg4){
            puts("a");
            continue;
        }
        flg1 = abs(pr - p1) <= r;
        flg2 = abs(pr - p2) <= r;
        flg3 = abs(pr - p3) <= r;
        if(flg1 || flg2 || flg3){
            puts("c");
            continue;
        }
        flg1 = dist_lc(pr, b, -a, -b * p1.real() + a * p1.imag()) <= r &&
               ip(p1-p2, pr-p2) >= 0 && ip(p2-p1, pr-p1) >= 0;
        flg2 = dist_lc(pr, d, -c, -d * p2.real() + b * p2.imag()) <= r &&
               ip(p2-p3, pr-p3) >= 0 && ip(p3-p2, pr-p2) >= 0;
        flg3 = dist_lc(pr, f, -e, -f * p3.real() + f * p3.imag()) <= r &&
               ip(p3-p1, pr-p1) >= 0 && ip(p1-p3, pr-p3) >= 0;
        if(flg1 || flg2 || flg3){
            puts("c");
            continue;
        }
        puts("d");
    }
    return 0;
}