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 <cstdio>
#include <cstdlib>
#include <cmath>
#include <climits>
#include <cfloat>
#include <map>
#include <utility>
#include <ctime>
#include <set>
#include <iostream>
#include <memory>
#include <string>
#include <cstring>
#include <vector>
#include <algorithm>
#include <functional>
#include <fstream>
#include <sstream>
#include <complex>
#include <stack>
#include <queue>
#include <cstring>
#include <numeric>
#include <cassert>
using namespace std;
static const double EPS = 1e-8;
typedef long long ll;
#define rep(i,n) for(int i=0;i<n;i++)
#define rev(i,n) for(int i=n-1;i>=0;i--)
#define all(a) a.begin(),a.end()
#define mp(a,b) make_pair(a,b)
#define pb(a) push_back(a)
#define SS stringstream
#define DBG1(a) rep(_X,sz(a)){printf("%d ",a[_X]);}puts("");
#define DBG2(a) rep(_X,sz(a)){rep(_Y,sz(a[_X]))printf("%d ",a[_X][_Y]);puts("");}
#define bitcount(b) __builtin_popcount(b)
#define EACH(i,c) for(typeof((c).begin()) i=(c).begin(); i!=(c).end(); ++i)

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

int chk(P a,P b,double r){
	double t = abs(a-b) - r;
	if( fabs(t) < EPS ) return 0;
	return t < 0 ? -1 : 1;
}
double cross(P a,P b){
	return imag(a*conj(b));
}
double dot(const P& a, const P& b) {return real(conj(a)*b);}



P projection(const L &l, const P &p) {double t = dot(p-l[0], l[0]-l[1]) / norm(l[0]-l[1]);return l[0] + t*(l[0]-l[1]);}

bool intersectSP(const L &s, const P &p) {return abs(s[0]-p)+abs(s[1]-p)-abs(s[1]-s[0]) < EPS;}

double distanceSP(const L &s, const P &p) {
	const P r = projection(s, p);
	if (intersectSP(s, r)) return abs(r - p);
	return min(abs(s[0] - p), abs(s[1] - p));
}

int main(){
	P a,b,c,d;
	double r;
	while(cin >> a.real() >> a.imag() , a.real() != 0 || a.imag() != 0 ){
		cin >> b.real() >> b.imag();
		cin >> c.real() >> c.imag();
		cin >> d.real() >> d.imag();
		cin >> r;
		
		int in = 0;
		int mk = cross(a-d,b-d) <= 0 ;
		if( mk == (cross(b-d,c-d) <= 0) && mk == (cross(c-d,a-d) <= 0)){
			in = 1;
		}
		
		if( chk(a,d,r) <= 0 &&  chk(b,d,r) <= 0 &&  chk(c,d,r) <= 0 ){
			cout << "b" << endl;
		}else if( chk(a,d,r) >= 0 &&  chk(b,d,r) >= 0 &&  chk(c,d,r) >= 0){
			if( distanceSP(L(a,b),d) < r - EPS || distanceSP(L(b,c),d) < r - EPS || distanceSP(L(a,c),d) < r - EPS  ){
				cout << "c" << endl;
			}else if( in ){
				cout << "a" << endl;
			}else if( distanceSP(L(a,b),d) > r + EPS || distanceSP(L(b,c),d) > r + EPS || distanceSP(L(a,c),d) > r + EPS )
				cout << "d" << endl;
			}
			cout << "c" << endl;
		}else{
			cout << "c" << endl;
		}
	

		
	}
}

C++: #include <iostream>
#include <vector>
#include <cmath>
#include <utility>
using namespace std;

#define x first
#define y second

typedef pair<double,double> pdd;
static const double EPS = 1e-9;

template<class T>
T sqr(T obj0) {
	return obj0 * obj0;
}

int is_cross(pdd l0, pdd l1, pdd p, double r) {
	pdd a,b,c;

	a.x = l0.x - p.x; a.y = l0.y - p.y;
	b.x = l1.x - p.x; b.y = l1.y - p.y;
	c.x = a.x - b.x; c.y = a.y - b.y;

	double d0, d1, d2, dist;
	double h;

	h = abs(a.x * b.y - a.y * b.x) / abs(sqrt(c.x * c.x + c.y * c.y));

	d0 = sqrt(sqr(c.x) + sqr(c.y));
	d1 = sqrt(sqr(a.x) + sqr(a.y) - sqr(h));
	d2 = sqrt(sqr(b.x) + sqr(b.y) - sqr(h));

	if(fabs(d0 - (d1 + d2)) < EPS)
		dist = h;
	else
		dist = min(sqrt(sqr(a.x) + sqr(a.y)), sqrt(sqr(b.x) + sqr(b.y)));

	if(fabs(dist - r) < EPS && dist <= r && r <= max(sqrt(sqr(a.x) + sqr(a.y)), sqrt(sqr(b.x) + sqr(b.y))))
		return 2;
	if(dist <= r && r <= max(sqrt(sqr(a.x) + sqr(a.y)), sqrt(sqr(b.x) + sqr(b.y))))
		return 1;
	return 0;
}

bool in_triangle(vector<pdd> tri, pdd p) {
	double s, t;
	pdd a, b, c;

	a.x = tri[1].x - tri[0].x; a.y = tri[1].y - tri[0].y;
	b.x = tri[2].x - tri[0].x; b.y = tri[2].y - tri[0].y;
	c.x = p.x - tri[0].x; c.y = p.y - tri[0].y;

	s = (c.x * b.y - c.y * b.x) / (a.x * b.y - a.y * b.x);
	t = (c.y * a.x - c.x * a.y) / (a.x * b.y - a.y * b.x);

	if(0 <= s + t && s + t <= 1 && 0 <= s && 0 <= t)
		return true;
	return false;
}

/*
bool in_circle(vector<pdd> t, pdd p, double r) {
	bool flg = true;
	for(int i = 0; i < 3; i++) {
		double tmp;
		tmp = sqr(t[i].x - p.x) + sqr(t[i].y - p.y);
		flg &= tmp <= r;
	}

	return flg;
}
*/

int main() {
	while(true) {
		vector<pdd> tri;
		pdd c;
		double r;

		for(int i = 0; i < 3; i++) {
			double tmpx, tmpy;
			cin >> tmpx >> tmpy;
			tri.push_back(pdd(tmpx, tmpy));
			if(i == 0 && tri[0].x == 0 && tri[0].y == 0)
				goto label;
		}
		cin >> c.x >> c.y >> r;

		bool flg0, flg1, flg2;
		flg0 = true;
		flg1 = false;
		flg2 = false;
		for(int i = 0; i < 3; i++) {
			flg0 &= ((sqr(c.x - tri[i].x) + sqr(c.y - tri[i].y)) <= r * r);
		}
		for(int i = 0; i < 3; i++) {
			int tmp;
			tmp = is_cross(tri[i % 3], tri[(i + 1) % 3], c, r);
			flg1 |= tmp;
			if(tmp == 2)
				flg2 = true;
		}

//		cout << boolalpha << flg0 << " " << flg1 << endl;

		if(flg0)
			cout << "b" << endl;
		else if((!flg1 || flg2)&& in_triangle(tri, c))
			cout << "a" << endl;
		else if(flg1)
			cout << "c" << endl;
		else
			cout << "d" << endl;
	}
label:;
}