Go: package main

import (
	"bufio"
	"fmt"
	"math"
	"os"
	"strconv"
	"strings"
)

var scanner = bufio.NewScanner(os.Stdin)

func main() {
	for scanner.Scan() {
		buf0 := strings.Split(scanner.Text(), " ")
		input0 := make([]int, 2)
		for i := range input0 {
			input0[i], _ = strconv.Atoi(buf0[i])
		}
		if input0[0] == 0 && input0[1] == 0 {
			break
		}
		a, b := input0[0], input0[1]
		scanner.Scan()
		buf1 := strings.Split(scanner.Text(), " ")
		input1 := make([]int, 2)
		for i := range input1 {
			input1[i], _ = strconv.Atoi(buf1[i])
		}
		c, d := input1[0], input1[1]
		scanner.Scan()
		buf2 := strings.Split(scanner.Text(), " ")
		input2 := make([]int, 2)
		for i := range input2 {
			input2[i], _ = strconv.Atoi(buf2[i])
		}
		e, f := input2[0], input2[1]
		scanner.Scan()
		buf3 := strings.Split(scanner.Text(), " ")
		input3 := make([]int, 2)
		for i := range input3 {
			input3[i], _ = strconv.Atoi(buf3[i])
		}
		g, h := input3[0], input3[1]
		scanner.Scan()
		r, _ := strconv.Atoi(scanner.Text())
		if (a-g)*(a-g)+(b-h)*(b-h) <= r*r && (c-g)*(c-g)+(d-h)*(d-h) <= r*r && (e-g)*(e-g)+(f-h)*(f-h) <= r*r {
			fmt.Println("b")
		} else if dist(a, b, c, d, g, h) > float64(r)-0.000001 && dist(a, b, e, f, g, h) > float64(r)-0.000001 && dist(e, f, c, d, g, h) > float64(r)-0.000001 && inclusive(a, b, c, d, e, f, g, h) {
			fmt.Println("a")
		} else if CL_intersect(a, b, c, d, g, h, r) || CL_intersect(a, b, e, f, g, h, r) || CL_intersect(e, f, c, d, g, h, r) {
			fmt.Println("c")
		} else {
			fmt.Println("d")
		}
	}
}

func CL_intersect(ax, ay, bx, by, cx, cy, r int) bool {
	acx := cx - ax
	acy := cy - ay
	bcx := cx - bx
	bcy := cy - by
	abx := bx - ax
	aby := by - ay
	if acx*acx+acy*acy >= r*r || bcx*bcx+bcy*bcy >= r*r {
		if abx*acx+aby*acy < 0 {
			if acx*acx+acy*acy <= r*r {
				return true
			}
		} else {
			if abx*acx+aby*acy > abx*abx+aby*aby {
				if bcx*bcx+bcy*bcy <= r*r {
					return true
				}
			} else if acx*acx+acy*acy-(abx*acx+aby*acy)*(abx*acx+aby*acy)/(abx*abx+aby*aby) <= r*r {
				return true
			}
		}
	}
	return false
}

func inclusive(ax, ay, bx, by, cx, cy, px, py int) bool {
	ok1 := true
	ok2 := true
	if (bx-ax)*(py-ay)-(px-ax)*(by-ay) < 0 {
		ok1 = false
	}
	if (bx-ax)*(py-ay)-(px-ax)*(by-ay) > 0 {
		ok2 = false
	}
	if (cx-bx)*(py-by)-(px-bx)*(cy-by) < 0 {
		ok1 = false
	}
	if (cx-bx)*(py-by)-(px-bx)*(cy-by) > 0 {
		ok2 = false
	}
	if (ax-cx)*(py-cy)-(px-cx)*(ay-cy) < 0 {
		ok1 = false
	}
	if (ax-cx)*(py-cy)-(px-cx)*(ay-cy) > 0 {
		ok2 = false
	}
	return ok1 || ok2
}

func dist(ax, ay, bx, by, px, py int) float64 {
	dx := float64(bx - ax)
	dy := float64(by - ay)
	A := float64(dx*dx + dy*dy)
	B := dx*float64(ax-px) + dy*float64(ay-py)
	t := -B / A
	if t < 0 {
		t = 0
	}
	if t > 1 {
		t = 1
	}
	tx := float64(ax) + dx*t
	ty := float64(ay) + dy*t
	return math.Sqrt((float64(px)-tx)*(float64(px)-tx) + (float64(py)-ty)*(float64(py)-ty))
}

C++: #include<iostream>
#include<stdio.h>
#include<cmath>

#define EPS 1E-5
#define SAME_SIGN( a , b ) a * b > EPS

namespace geometry{

	struct point{
		double x;
		double y;
		point(){}
		point( double x , double y ):x(x),y(y){}
		point& operator += ( const point& p ){ x += p.x; y += p.y; return *this; }
		point& operator -= ( const point& p ){ x -= p.x; y -= p.y; return *this; }
		point& operator *= ( double factor ){ x *= factor; y *= factor; return *this; }
		point& operator /= ( double factor ){ x /= factor; y /= factor; return *this; }
		bool operator == ( const point& p ) const { return (std::abs(x - p.x) < EPS && std::abs(y - p.y) < EPS); }
		bool operator != ( const point& p ) const { return !(*this == p); }
		double norm() const{ return sqrt( pow( x , 2) + pow( y , 2 ) ); }
	};

	typedef point vector2d;

	struct circle{
		point center;
		double radius;
	};

	struct triangle{
		point a;
		point b;
		point c;
	};

	point operator + ( const point& p1 , const point& p2 ){
			point p(p1);
			p += p2;
			return p;
		}
		point operator - ( const point& p1 , const point& p2 ){
			point p(p1);
			p -= p2;
			return p;
		}
		point operator * ( const point& p1 , double factor ){
			point p(p1);
			p *= factor;
			return p;
		}
		point operator * ( double factor , const point& p1 ){
			point p(p1);
			p *= factor;
			return p;
		}
		point operator / ( const point& p1 , double factor ){
			point p(p1);
			p /= factor;
			return p;
		}

	std::istream& operator >> ( std::istream& s , point& p )
	{
		return s >> p.x >> p.y;
	}

	double get_distance( const point& p1 , const point& p2 )
	{
		return sqrt( pow( p1.x - p2.x , 2 ) + pow( p1.y - p2.y , 2 ) );
	}

	double get_dot_product( const point& a , const point& b , const point& o )
	{
		vector2d oa = a - o;
		vector2d ob = b - o;
		return oa.x * ob.x + oa.y * ob.y;
	}

	double get_cross_product( const point& a , const point& b , const point& o )
	{
		vector2d oa = a - o;
		vector2d ob = b - o;
		return oa.x * ob.y - oa.y * ob.x;
	}

	point get_project( const point& s , const point& e , const point& p ){
		double scale = get_dot_product( s , e , p ) / ( e - s ).norm();
		return s + scale * ( e - s ) / ( e - s ).norm();
	}

	bool is_inter_point( const point& s , const point& e , const point& p ){
		return std::abs(get_distance( s , p ) + get_distance( e , p ) - get_distance( s , e )) < EPS;
	}
}

using namespace geometry;

bool input_data( triangle& tri , circle& cir )
{
	std::cin >> tri.a;
	if( tri.a.x == 0 && tri.a.y == 0 ){
		return false;
	}
	std::cin >> tri.b;
	if( tri.b.x == 0 && tri.b.y == 0 ){
		return false;
	}
	std::cin >> tri.c;
	if( tri.c.x == 0 && tri.c.y == 0 ){
		return false;
	}
	std::cin >> cir.center;
	if( cir.center.x == 0 && cir.center.y == 0 ){
		return false;
	}
	std::cin >> cir.radius;
	return true;
}

bool is_overlap_for_one_side( const point& p1 , const point& p2 , circle& cir )
{
	if( is_inter_point( p1 , p2 , get_project( p1 , p2 , cir.center )) ){
		if( get_distance( get_project( p1 , p2 , cir.center ) , cir.center ) <= cir.radius + EPS ){
			return true;
		}
		else{
			return false;
		}
	}
	else{
		return false;
	}
}

bool is_overlap( triangle& tri , circle& cir )
{
	// オーバーラップなし、あるいは円が三角形に完全に含まれる場合にfalse
	if( !(is_overlap_for_one_side( tri.a , tri.b , cir )) &&
		!(is_overlap_for_one_side( tri.b , tri.c , cir )) &&
		!(is_overlap_for_one_side( tri.c , tri.a , cir ))
		){
			return false;
	}
	else{
		return true;
	}

}

void output_positional_relation_of_circle_and_triangle( triangle& tri , circle& cir )
{
	if( get_distance( tri.a , cir.center ) <= cir.radius + EPS &&
		get_distance( tri.b , cir.center ) <= cir.radius + EPS &&
		get_distance( tri.c , cir.center ) <= cir.radius + EPS
		){
			std::cout << 'b' << std::endl;
	}
	else{
		if( get_distance( tri.a , cir.center ) > cir.radius + EPS &&
			get_distance( tri.b , cir.center ) > cir.radius + EPS &&
			get_distance( tri.c , cir.center ) > cir.radius + EPS
			){
				if( !(is_overlap( tri , cir )) ){
					if( SAME_SIGN( get_cross_product( tri.a , tri.b , cir.center ) , get_cross_product( tri.b , tri.c , cir.center ) ) &&
						SAME_SIGN( get_cross_product( tri.b , tri.c , cir.center ) , get_cross_product( tri.c , tri.a , cir.center ) )
						){
							std::cout << 'a' << std::endl;
					}
					else{
						std::cout << 'd' << std::endl;
					}
				}
				else{
					std::cout << 'c' << std::endl;
				}
		}
		else{
			std::cout << 'c' << std::endl;
		}
	}
}

int main(void)
{
	triangle tri;
	circle cir;

	while(input_data(tri, cir)){
		output_positional_relation_of_circle_and_triangle( tri , cir );
	}

	return 0;
}