#include <cstdio>
#include <complex>
#include <vector>
#include <algorithm>
#include <cmath>
using namespace std;

#define EPS 1e-6
typedef complex<double> P;

struct L : public vector<P> {
  L(const P &a, const P &b) {
    push_back(a); push_back(b);
  }
};

struct C {
  P p; double r;
  C(const P &p, double r) : p(p), r(r) { }
};

double cross(const P& a, const P& b) {
  return imag(conj(a)*b);
}

double dot(const P& a, const P& b) {
  return real(conj(a)*b);
}

double dist(const P& a, const P& b)
{
	return sqrt((a.real()-b.real())*(a.real()-b.real())+(a.imag()-b.imag())*(a.imag()-b.imag()));
}

bool isIntoC(const P& p, const C& c)
{
	return dist(p, c.p) < c.r+EPS;
}

bool isIntoC(const L& l, const C& c)
{
	return dist(l[0], c.p) < c.r+EPS && dist(l[1], c.p) < c.r+EPS;
}

int CrossPtNum(const C& c, const L& l)
{
	P LV=l[0]-l[1];
	double A=(LV.real()*LV.real())+(LV.imag()*LV.imag());
	double B=(LV.real()*l[0].real())-(LV.real()*c.p.real())+(LV.imag()*l[0].imag())-(LV.imag()*c.p.imag());
	double C=(l[0].real()-c.p.real())*(l[0].real()-c.p.real())+(l[0].imag()-c.p.imag())*(l[0].imag()-c.p.imag())-(c.r*c.r);

	if(fabs(B*B-A*C) < EPS) return 1;
	if(B*B-A*C  > 0) return 2;
	
	return 0;
}

bool intersectSC(const L& l, const C& c)
{
	if(isIntoC(l, c)) return false;
	
	double vc=dot(l[1]-l[0], c.p-l[0]);
	if(vc<0)
	{
		double dif=c.r-dist(l[0], c.p);
		return dif+EPS > 0;
	}
	
	double vv=dot(l[1]-l[0], l[1]-l[0]);
	
	if(vc > vv)
	{
		double d=dist(l[1], c.p);
		double dif=c.r*c.r-d*d;
		
		return dif+EPS > 0;
	}
	
	double cc=dot(c.p-l[0], c.p-l[0]);
	double dif=c.r*c.r-(cc-(vc/vv)*vc);
	
	return dif+EPS > 0;
}

int main()
{
	int x,y,r;
	while(scanf("%d%d", &x, &y), (x||y))
	{
		vector<P> tri;
		for(int i=0; i<3; i++)
		{
			tri.push_back(P(x,y));
			if(i==2) break;
			scanf("%d%d", &x, &y);
		}
	
		scanf("%d%d%d", &x,&y,&r);
		C c(P(x,y), r);
		
		bool g=true;
		for(int i=0; i<3; i++)
		{
			if(!isIntoC(tri[i], c))
			{
				g=false;
				break;
			}
		}
		
		if(g) 
		{
			puts("b");
			continue;
		}
		int cr[3];
		cr[0]=cross(tri[0]-tri[1], tri[0]-c.p);
		cr[1]=cross(tri[1]-tri[2], tri[1]-c.p);
		cr[2]=cross(tri[2]-tri[0], tri[2]-c.p);
		
		if(cr[0]<EPS&&cr[1]<EPS&&cr[2]<EPS||cr[0]+EPS>0&&cr[1]+EPS>0&&cr[2]+EPS>0)
		{
			g=true;
			for(int i=0; i<3; i++)
			{
				if(CrossPtNum(c, L(tri[i], tri[(i+1)%3])) == 2) g=false;
			}
			if(g) 
			{
				puts("a");
				continue;
			}
			
		}
		
		for(int i=0; i<tri.size(); i++)
		for(int j=i+1; j<tri.size(); j++)
		{
			if(intersectSC(L(tri[i], tri[j]), c))	g=true;
		}
		if(g)
		{
			puts("c");
			continue;
			
		}
		
		puts("d");
	}
}