#include <iostream>
#include <cmath>
#include <vector>
#include <algorithm>

#define REP(i,k,n) for(int i=k;i<n;i++)
#define rep(i,n) for(int i=0;i<n;i++)
#define EPS 1e-10

using namespace std;

struct Point 
{
    double x, y;
    Point(double x=0, double y=0) : x(x), y(y) {}
    
	double distance (const Point &o) const 
	{
        return sqrt((x - o.x) * (x - o.x) + (y - o.y) * (y - o.y));
    }

    Point operator+(const Point &o) const 
	{
        return Point(x+o.x, y+o.y);
    }
 
    Point operator-(const Point &o) const 
	{
        return Point(x-o.x, y-o.y);
    }
 	
    Point operator*(const double m) const 
	{
   	   return Point(x*m, y*m);
    }

    Point operator/(const double d) const 
	{
       return Point(x/d, y/d);
    }

    double cross(const Point &o) const 
	{
       return x * o.y - y * o.x;
    }
 
    double dot(const Point &o) const 
	{
        return x * o.x + y * o.y;
    }
 
    double atan() const 
	{
        return atan2(y, x);
    }

    double norm() const 
	{
        return sqrt(dot(*this));
    }

	double area_abs(const Point &a,const Point &b) const
	{
		double t = (a.x - x) * (b.y - y);
		double t2 = (a.y - y) * (b.x - x);
		double areas = abs(t-t2);
		return areas/2;
	}

	//符号付き面積
	double area(const Point &a,const Point &b)
	{
		return ((a.x - x) * (b.y - y)) + ((b.x - x) * (y - a.y));
	}
	//線分abが自身の含まれているのかどうか判断する
	int between(const Point &a,const Point &b)
	{
		if(area(a,b) != 0) return 0;

		if(a.x != b.x)
		{
			return ((a.x <= x) && (x <= b.x) || (a.x >= x) && (x >= b.x));
		}
		else
		{
			return ((a.y <= y) && (y <= b.y) || (a.y >= y) && (y >= b.y));
		}
	}

	double distance_seg(const Point& a,const Point& b)
	{
		if((b-a).dot(*this-a) < EPS) 
		{
			return (*this-a).norm();
		}
		if((a-b).dot(*this-b) < EPS)
		{
			return (*this-b).norm();
		}
		return abs((b-a).cross(*this-a)) / (b-a).norm();
	}

	//点a,b,cからなる三角形の中に自身が含まれているかどうか
	bool hitPolygon(const Point& a,const Point& b,const Point& c)
	{
		double t = (b-a).cross(*this-b);
		double t2 = (c-b).cross(*this-c);
		double t3 = (a-c).cross(*this-a);
		
		if((t > 0 && t2 > 0 && t3 > 0) || ( t < 0 && t2 < 0 && t3 < 0))
		{
			return 1;
		}

		return 0;
	}
};

int main()
{
	double a,b;
	while(cin >> a >> b)
	{
		if(a == 0 && b == 0) break;
		
		vector<Point> t;
		t.push_back(Point(a,b));

		rep(i,2)
		{
			cin >> a >> b;
			t.push_back(Point(a,b));
		}

		double r;
		cin >> a >> b >> r;
		Point c(a,b);

		int cnt = 0;

		rep(i,3)
		{
			if(t[i].distance(c) > r)
			{
				cnt++;
			}
		}

		if(cnt == 0) cout << "b" << endl;
		else if(cnt == 1 || cnt == 2) cout << "c" << endl;
		else
		{
			bool flag = false;

			if(c.distance_seg(t[0],t[1]) < r) flag = true;
			if(c.distance_seg(t[1],t[2]) < r) flag = true;
			if(c.distance_seg(t[2],t[0]) < r) flag = true;
			
			if(flag) cout << "c" << endl;
			else
			{
				if(c.hitPolygon(t[0],t[1],t[2]))
				{
					cout << "a" << endl;
				}
				else
				{
					bool flag2 = false;

					if(fabs(c.distance_seg(t[0],t[1])-r) < EPS) flag2 = true;
					if(fabs(c.distance_seg(t[1],t[2])-r) < EPS) flag2 = true;
					if(fabs(c.distance_seg(t[2],t[0])-r) < EPS) flag2 = true;

					if(flag2) cout << "c" << endl;
					else cout << "d" << endl;
				}
			}
		}
	}
	return 0;
}

Measured memory_bytes: 1216