#include <iostream>
#include <sstream>
#include <string>
#include <algorithm>
#include <vector>
#include <map>
#include <set>
#include <bitset>
#include <queue>
#include <stack>
#include <utility>
#include <complex>
#include <cstring>
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <cctype>
using namespace std;
 
#define rep(i,n) for(int i=0; i<n; i++)
#define repa(i,s,e) for(int i=s; i<=e; i++)
#define repd(i,s,e) for(int i=s; i>=e; i--)

typedef vector<int> vi;
typedef long long ll;
typedef pair<ll,ll> pll;

typedef complex<ll> P;

int x, y, r;
P tri[3];
pair<P,ll> circle;

ll cross(P e1, P e2) {
	return e1.real() * e2.imag() - e1.imag() * e2.real();
}

bool inc() {
	P e0 = tri[0] - circle.first;
	P e1 = tri[1] - circle.first;
	P e2 = tri[2] - circle.first;
	ll d0 = cross(e0, e1);
	ll d1 = cross(e1, e2);
	ll d2 = cross(e2, e0);
	if(d0 * d1 > 0 && d1 * d2 > 0 && d2 * d0 > 0) return true;
	return false;
}

ll norm2(P p) {
	return p.real() * p.real() + p.imag() * p.imag();
}

bool proj(P p1, P p2, P p) {
	P e12 = p2 - p1;
	P e1 = p - p1;
	P e2 = p2 - p;
	int d1 = e1.real() * e12.real() + e1.imag() * e12.imag();
	int d2 = e2.real() * e12.real() + e2.imag() * e12.imag();
	if(d1 >= 0 && d2 >= 0) return true;
	return false;
}

void dist2(P p1, P p2, P p, ll& nm, ll& de) {
	if(!proj(p1, p2, p)) {
		nm = min(norm2(p1 - p) , norm2(p2 - p));
		de = 1;
		return;
	}

	ll a = p2.imag() - p1.imag();
	ll b = -(p2.real() - p1.real());
	ll c = - a * p1.real() - b * p1.imag();
	nm = a * p.real() + b * p.imag() + c;
	nm = nm * nm;
	de = a * a + b * b;
	if(de == 0) {
		nm = norm2(p1 - p);
		de = 1;
	}
}


void solve() {
	ll r2 = circle.second * circle.second;
	vector<pll> d(6);
	d[3].first = norm2(tri[0] - circle.first); d[3].second = 1LL;
	d[4].first = norm2(tri[1] - circle.first); d[4].second = 1LL;
	d[5].first = norm2(tri[2] - circle.first); d[5].second = 1LL;
	if(d[3].first <= r2 && d[4].first <= r2 && d[5].first <= r2) {
		cout << "b" << endl;
		return;
	}

	bool iscin = inc();
	dist2(tri[0], tri[1], circle.first, d[0].first, d[0].second);
	dist2(tri[1], tri[2], circle.first, d[1].first, d[1].second);
	dist2(tri[2], tri[0], circle.first, d[2].first, d[2].second);
	if(iscin && (d[0].first >= r2 * d[0].second) &&
		        (d[1].first >= r2 * d[1].second) &&
		        (d[2].first >= r2 * d[2].second)) {
		cout << "a" << endl;
		return;
	}

	bool inter = false;
	rep(i,6) {
		if(d[i].first <= r2 * d[i].second) {
			inter = true;
			break;
		}
	}

	if(inter) {
		cout << "c" << endl;
		return;
	}

	cout << "d" << endl;
}

int main() {
	while( 1 ) {
		scanf("%d%d", &x, &y);
		if(x == 0 && y == 0) break;
		tri[0] = P(x, y);

		scanf("%d%d", &x, &y);
		tri[1] = P(x, y);
		
		scanf("%d%d", &x, &y);
		tri[2] = P(x, y);
		
		scanf("%d%d", &x, &y);
		scanf("%d", &r);
		circle.first = P(x, y);
		circle.second = r;

		solve();
	}
}