#include<iostream>
#include<cmath>
#include<cstdlib>

using namespace std;

struct Point{ int x,y; };
struct Triangle{ Point a,b,c; };
struct Circle{ Point c; int r; };

inline double dist(const Point &p1, const Point &p2){
	return sqrt( pow(p2.x-p1.x,2.0)+pow(p2.y-p1.y,2.0) );
}
double dist(const Point &p1, const Point &p2, const Point &p)
{
	if( p1.x == p2.x ){
		return abs( p1.x - p.x );
	}else{
		double a = -(p2.y-p1.y)/double(p2.x-p1.x);
		double b = 1;
		double c = -a * p1.x - p1.y;

		return fabs( a * p.x + b * p.y + c ) / sqrt( pow(a,2.0)+pow(b,2.0) );
	}
}
inline bool isLeft(const Point &a, const Point &b, const Point &p){
	double vx = b.x - a.x, vy = b.y - a.y;
	double px = p.x - a.x, py = p.y - a.y;
	return (vx * py - px * vy) > 0;
}

inline bool isInside(const Point &p, const Triangle &t )
{
	if( !isLeft( t.a, t.b, p ) )
		return false;
	if( !isLeft( t.b, t.c, p ) )
		return false;
	if( !isLeft( t.c, t.a, p ) )
		return false;

	return true;
}

bool Contains(const Triangle &t, const Circle &c)
{
	if( isInside( c.c, t ) ){
		if( c.r < dist( t.a, t.b, c.c ) &&
			c.r < dist( t.b, t.c, c.c ) &&
			c.r < dist( t.c, t.a, c.c ) )
			return true;
	}
	return false;
}

bool IsContained(const Triangle &t, const Circle &c)
{
  if( dist( t.a, c.c ) < c.r &&
	  dist( t.b, c.c ) < c.r &&
	  dist( t.c, c.c ) < c.r )
	  return true;
  else
	  return false;
}
bool isCross(const Point &tp1, const Point &tp2, const Circle &c)
{
	Point p1 = tp1;
	Point p2 = tp2;
	if( p2.x < p1.x )
		swap( p1, p2 );

	if( dist( p1, p2, c.c ) > c.r )
		return false;
	else{
		if( p1.x != p2.x ){
			double x1, x2;
			double a = (p2.y - p1.y) / (p2.x - p1.x);
			double b = -a * p1.x + p1.y;
			double A = 1 + pow(a,2.0);
			double B = -( c.c.x - a * c.c.x + a * c.c.y );
			double C = pow( c.c.x, 2.0 ) + pow( b, 2.0 ) - 2*b*c.c.y + pow( c.c.y, 2.0 ) - pow( c.r,2.0 );

			x1 = ( -B + sqrt( pow(B,2.0) - A * C ) ) / A;
			x2 = ( -B - sqrt( pow(B,2.0) - A * C ) ) / A;

			if( p1.x <= x1 && x1 <= p2.x || p1.x <= x2 && x2 <= p2.x )
				return true;
			else
				return false;
		}else{
			if( p2.y < p1.y )
				swap( p1, p2 );

			double d = dist( p1, p2, c.c );
			double l = sqrt( pow(c.r,2.0) - pow(d,2.0) );

			double y1 = c.c.y - l;
			double y2 = c.c.y + l;

			if( p1.y <= y1 && y1 <= p2.y || p1.y <= y2 && y2 <= p2.y )
				return true;
			else
				return false;
		}
	}
}
bool isCross(const Triangle &t, const Circle &c ){

	if( !isCross(t.a,t.b,c) && !isCross(t.b,t.c,c) && !isCross(t.c,t.a,c) )
		return false;
	else
		return true;
}

char judgeRel(const Triangle &t, const Circle &c){
	if( Contains( t,c ) )
		return 'a';
	if( IsContained( t, c ) )
		return 'b';
	if( !isCross( t,c ) )
		return 'd';
	else
		return 'c';
}

int main()
{
	while(true){
		Triangle t;
		Circle c;
		cin >> t.a.x >> t.a.y;
		if( t.a.x == 0 && t.a.y == 0 )
			break;
		cin >> t.b.x >> t.b.y >> t.c.x >> t.c.y >> c.c.x >> c.c.y >> c.r;
		cout << judgeRel( t, c ) << endl;
	}
	return 0;
}