#include <bits/stdc++.h>
 
using namespace std;
 
#define EPS 1e-7
 
class Point {
  public:
    double x,y;
    Point(double x = 0.0, double y = 0.0) : x(x), y(y) {}
    Point operator + (Point p) { return Point(x+p.x, y+p.y); }
    Point operator - (Point p) { return Point(x-p.x, y-p.y); }
    Point operator * (double a) { return Point(x*a, y*a); }
    Point operator / (double a) { return Point(x/a, y/a); }
    double abs() { return sqrt(norm()); }
    double norm() { return x * x + y * y; }
};
 
double dot(Point a, Point b)
{
    return a.x*b.x + a.y*b.y;
}
double cross(Point a, Point b)
{
    return a.x*b.y - a.y*b.x;
}
 
class Segment {
  public:
    Point t, s;
    Segment(Point t, Point s) : t(t), s(s) {}
};
 
class Circle {
  public:
    double x, y, r;
    Circle(double x = 0.0, double y = 0.0, double r = 0.0) : x(x), y(y), r(r) {}
};
 
Point point[3];
Circle ci;
 
int ccw(Point p0, Point p1, Point p2)
{
    Point a = p1 - p0;
    Point b = p2 - p0;
    if (cross(a, b) > EPS) return 1;
    if (cross(a, b) < -EPS) return -1;
    if (dot(a, b) < -EPS) return 2;
    if (a.norm() < b.norm()) return -2;
    return 0;
}
 
bool check()
{
    Point cp = Point(ci.x, ci.y);
    int c1 = ccw(point[0], point[1], cp);
    int c2 = ccw(point[1], point[2], cp);
    int c3 = ccw(point[2], point[0], cp);
    return (c1 == c2 && c2 == c3);
}
 
typedef Segment Line;

double getDistanceLP(Line l, Point p)
{
    return abs(cross(l.s-l.t, p-l.t) / (l.s-l.t).abs());
}
 
double getDistanceSP(Segment seg, Point p)
{
    if (dot(seg.s-seg.t, p-seg.t) < 0.0) return (p-seg.t).abs();
    if (dot(seg.t-seg.s, p-seg.s) < 0.0) return (p-seg.s).abs(); 
    return getDistanceLP(seg, p);
}
 
int getStateCS(Circle c, Segment seg)
{
    Point cp = Point(c.x, c.y);
    if (getDistanceSP(seg, cp) >= c.r) {
	return 1;
    }
    return 3;
}
 
bool isCross()
{
    Point cp = Point(ci.x, ci.y);
 
    Segment seg[3] = {
	Segment(point[0], point[1]),
	Segment(point[1], point[2]),
	Segment(point[2], point[0])
    };
    int cnt = 0;
    for (int i = 0; i < 3; i++) {
	if (getDistanceSP(seg[i], cp) > ci.r) {
	    cnt++;
	}
    }
 
    if (cnt == 3 && !check()) return false;
    return true;
}
 
void judge()
{
    Segment seg[3] = {
	Segment(point[0], point[1]),
	Segment(point[1], point[2]),
	Segment(point[2], point[0])
    };
    Point cp = Point(ci.x, ci.y);
    int TinC = 0;
    for (int i = 0; i < 3; i++, TinC++) {
	if ((point[i]-cp).abs() > ci.r) {
	    break;
	}
    }
 
    if (TinC == 3) { 
	cout << 'b' << endl;
	return;
    }
 
    int CinT = 0;
    for (int i = 0; i < 3; i++) {
	if (getStateCS(ci, seg[i]) == 1) {
	    CinT++;
	}
    }
    if (CinT == 3 && check()) {
	cout << 'a' << endl;
    } else {
	if (isCross()) {
	    cout << 'c' << endl;
	} else {
	    cout << 'd' << endl;
	}
    }
}
 
int main()
{
    while (1) {
	cin >> point[0].x >> point[0].y;
	if (!point[0].x && !point[0].y) break;
	for (int i = 0; i < 2; i++) {
	    cin >> point[i+1].x >> point[i+1].y;
	}
	cin >> ci.x >> ci.y >> ci.r;
	judge();
    }
    return 0;
}