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 <bits/stdc++.h>
using namespace std;

typedef complex<double> P;

P in(){
	double a,b;
	cin >> a >> b;
	return P(a,b);
}
double dot(P a,P b){
	return (conj(a)*b).real();	
}
double cross(P a,P b){
	return (conj(a)*b).imag();
}

// a-b vs c
P pedal(P a,P b,P c){
	P v = b - a;
	P e = v / abs(v);
	P p = c - a;
	return a + e * dot(e,p);
}


double eps = 1e-9;
bool intersectSP(P a,P b,P c){
	return fabs( abs(a-b) - (abs(a-c) + abs(b-c)) ) < eps;
}



double distanceSP(P a,P b,P c){
	P pdl = pedal(a,b,c);
	double answer = min(abs(a-c),abs(b-c));
	if( intersectSP(a,b,pdl) ) answer = min(answer, abs(pdl-c) );
	return answer;
}


/*
a / a(a) * b
 
      a
----*------->
 .  |
 b\ |
   \|

*/
int main(){

	while(true){
		vector<P> p(3);
		for(int i = 0 ; i < 3 ; i++) p[i] = in();
	
		if( abs(p[0]) < eps ) return 0; // end
		P d = in();

		double R;
		cin >> R;
		for(int i = 0 ; i < 3 ; i++) p[i] -= d;

		vector<int> sgn;
		for(int i = 0 ; i < 3 ; i++){
			double s = cross(p[i],p[(i+1)%3]);
			if( s < 0 ) s = -1;
			else if( s > 0 ) s = +1;
			if(s) sgn.push_back(s);
		}
		bool isin = false;
		if( sgn.size() == 0 || count(sgn.begin(),sgn.end(),sgn[0]) == 3 ) isin = true;
		double dist = 1e9;

		for(int i = 0 ; i < 3 ; i++)
			dist = min(distanceSP(p[i],p[(i+1)%3],P(0,0)),dist);
		if( isin && dist > R - eps ){
			cout << "a" << endl;
		}else if( max( {abs(p[0]),abs(p[1]),abs(p[2])}) < R - eps ){
			cout << "b" << endl;
		}else if( !isin && dist > R + eps ){
			cout << "d" << endl;
		}else{
			cout << "c" << endl;
		}
	}



}

C++: #include<iostream>
#include<cmath>
#include<cstdlib>
#include<cassert>

#define SQRD(a) ((a)*(a))

using namespace std;

struct Point{ double 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) );
}
inline 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 c = -a * p1.x - p1.y;

		return fabs( a * p.x + p.y + c ) / sqrt( SQRD(a) + 1 );
	}
}
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 isRight(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 ) && isLeft( t.b, t.c, p ) && isLeft( t.c, t.a, p ) )
		return true;
	if( isRight( t.a, t.b, p ) && isRight( t.b, t.c, p ) && isRight( t.c, t.a, p ) )
		return true;
	return false;
}

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 &p1, const Point &p2, const Circle &c)
{
	Point a1;
	Point a2;
	double d = dist(p1,p2,c.c);
	double l = dist(p1,p2);
			
	if( d > c.r )
		return false;
	else{
		if( p1.x != p2.x ){
			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 = a * b - c.c.x - a * c.c.y;
			double C = SQRD(c.c.x) + SQRD(c.c.y) - SQRD(c.r) + SQRD(b) - 2*b*c.c.y;

			assert( SQRD(B)-A*C >= 0 );
			a1.x = ( -B + sqrt( SQRD(B) - A * C ) ) / A;
			a2.x = ( -B - sqrt( SQRD(B) - A * C ) ) / A;
			a1.y = a * a1.x + b;
			a2.y = a * a1.x + b;
			
			if( dist(a1,p1) <= l && dist(a1,p2) <= l ||
				dist(a2,p1) <= l && dist(a2,p2) <= l )
				return true;
			else
				return false;
		}else{
			double e = sqrt( SQRD(c.r) - SQRD(d) );

			a1.x = p1.x; a1.y = c.c.y - e;
			a2.x = p2.x; a2.y = c.c.y + e;

			if( dist(a1,p1) <= l && dist(a1,p2) <= l ||
				dist(a2,p1) <= l && dist(a2,p2) <= l )
				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 true;
	else
		return false;
}

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 'c';
	else
		return 'd';
}

int main()
{
	while(true){
		Triangle t;
		Circle c;

		cin >> t.a.x >> t.a.y;
		if( cin.eof() )
			break;
		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;
}