#include <iostream>
#include <cstdio>
#include <cmath>
#include <vector>
#include <map>
#include <stack>
#include <queue>
#include <algorithm>
#include <set>

#define FOR(i,a,b) for(int i=(a);i<(b);i++)
#define REP(i,j) FOR(i,0,j)
#define mp std::make_pair

typedef long long ll;
typedef unsigned long long ull;

const int INF = 1001001001;

// S N E W(南北東西)
const int dx[8] = {0, 0, 1, -1, 1, 1, -1, -1}, dy[8] = {1, -1, 0, 0, 1, -1, 1, -1};

struct P{
    P(){}
    P(ll _r, ll _i):r(_r), i(_i){}
    void real(const ll& v){r = v;}
    void imag(const ll& v){i = v;}
    ll r, i;
};

ll real(const P& p){return p.r;}
ll imag(const P& p){return p.i;}
ll norm(const P& p){ll r = real(p), i = imag(p); return r * r + i * i;}

P operator+(const P& lhs, const P& rhs){
    return P(real(lhs)+real(rhs), imag(lhs)+imag(rhs));
}
P operator-(const P& lhs, const P& rhs){
    return P(real(lhs)-real(rhs), imag(lhs)-imag(rhs));
}
P operator-(const P& p){
    return P(-real(p), -imag(p));
}

ll cross(const P& lhs, const P& rhs){
    return real(lhs)*imag(rhs) - imag(lhs)*real(rhs);
}

ll dot(const P& lhs, const P& rhs){
    return real(lhs)*real(rhs) + imag(lhs)*imag(rhs);
}

// (distance between ab and p) ^ 2
ll distance(const P& a, const P& b, const P& p){
    ll c = cross(p-a, b-a);
    return c * c / norm(b-a);
}

P ps[3], cc;
ll radius;

bool doesLineAcrossCircle(const P& a, const P& b){
    // printf("(%lld, %lld)-(%lld, %lld): %lld\n",
    //        real(a), imag(a), real(b), imag(b), distance(a, b, cc));
    if(distance(a, b, cc) > radius * radius){return false;}
    if(dot(cc-a, b-a) * dot(cc-b, b-a) > 0){return false;}
    return true;
}

bool isCircleInTriangle(){
    REP(i, 3){
        if(distance(ps[i], ps[(i+1)%3], cc) < radius * radius){return false;}
    }

    int count_n = 0, count_p = 0;
    REP(i, 3){
        ll c = cross(cc-ps[i], ps[(i+1)%3]-ps[i]);
        if(c < 0){
            count_n += 1;
        }else if(c == 0){
            count_n += 1; count_p += 1;
        }else{
            count_p += 1;
        }
    }

    if(count_n % 3 == 0 || count_p % 3 == 0){return true;}
    return false;
}

bool isTriangleInCircle(){
    REP(i, 3){
        if(norm(ps[i]-cc) > radius * radius){return false;}
    }
    return true;
}

bool isIntersected(){
    REP(i, 3){
        if(doesLineAcrossCircle(ps[i], ps[(i+1)%3])){return true;}
    }
    return false;
}

int main(){
    ll re, im;
    while(std::cin >> re >> im, re || im){
        ps[0].real(re); ps[0].imag(im);
        
        FOR(i, 1, 3){
            ll re, im;
            std::cin >> re >> im;
        
            ps[i].real(re); ps[i].imag(im);
        }
        
        {
            ll re, im;
            std::cin >> re >> im;
            
            cc.real(re); cc.imag(im);
        }
        
        std::cin >> radius;
        
        if(isCircleInTriangle()){puts("a");}
        else if(isTriangleInCircle()){puts("b");}
        else if(isIntersected()){puts("c");}
        else{puts("d");}
    }   
}