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;

#define FOR(i,a,b) for(int i=(a);i<(b);i++)
#define REP(i,n) FOR(i,0,n)
#define EPS (1e-8)

typedef complex<double> P;
typedef P Vec;

struct Line {
	P p;
	Vec v;
	
	Line() {}
	Line(P p, Vec v) : p(p), v(v) {}
	
	P getPoint(double t) {
		return p + v * t;
	}
};

struct Circle {
	P p;
	double r;
	
	Circle() {}
	Circle(P p, double r) : p(p), r(r) {}
};

double dot(Vec a, Vec b) {
	return a.real() * b.real() + a.imag() * b.imag();
}

double cross(Vec a, Vec b) {
	return a.real() * b.imag() - a.imag() * b.real();
}

bool isInCir(Circle &c, P &p) {
	return norm(c.p - p) <= c.r*c.r + EPS;
}

bool isInSeg(Line &s, P &p) {
	Vec a = s.v, b = p - s.p;
	double l1 = abs(a), l2 = abs(b);
	
	return dot(a, b) >= l1*l2 - EPS && dot(a, b) <= l1*l2 + EPS && l1 >= l2 - EPS;
}

bool isInTri(P &a, P &b, P &c, P &d) {
	P p[4];
	p[0] = a; p[1] = b; p[2] = c; p[3] = d;
	
	int cnt = 0;
	REP(i, 3) {
		Vec v = p[3] - p[i];
		Line s(p[i], p[(i + 1) % 3] - p[i]);
		
		if (isInSeg(s, p[3])) return true;
		if (cross(s.v, v) < -EPS) cnt++;
	}
	return (cnt == 0 || cnt == 3);
}

double distanceSP(Line &s, P &p) {
	Vec a = s.v, b = p - s.p;
	return abs(cross(a, b)) / abs(a);
}

int main() {
	double x1, y1, x2, y2, x3, y3, x4, y4, r;
	while (cin >> x1 >> y1, x1 || y1) {
		cin >> x2 >> y2 >> x3 >> y3 >> x4 >> y4 >> r;
		
		P p[4] = {P(x1, y1), P(x2, y2), P(x3, y3), P(x4, y4)};
		Line segment[3] = {Line(p[0], p[1] - p[0]), Line(p[1], p[2] - p[1]), Line(p[2], p[0] - p[2])};
		Circle c(p[3], r);
		
		if ( isInTri(p[0], p[1], p[2], p[3]) &&
			 distanceSP(segment[0], c.p) >= c.r - EPS &&
			 distanceSP(segment[1], c.p) >= c.r - EPS &&
			 distanceSP(segment[2], c.p) >= c.r - EPS )
			puts("a");
		else if ( isInCir(c, p[0]) && isInCir(c, p[1]) && isInCir(c, p[2]) )
			puts("b");
		else if ( isInCir(c, p[0]) || isInCir(c, p[1]) || isInCir(c, p[2]) )
			puts("c");
		else
			puts("d");
	}
	return 0;
}

C++: #include<stdio.h>
#include<algorithm>
using namespace std;
bool CL_intersect(int ax,int ay,int bx,int by,int cx,int cy,int r){
	int acx=cx-ax;
	int acy=cy-ay;
	int bcx=cx-bx;
	int bcy=cy-by;
	int abx=bx-ax;
	int 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-(long long)(abx*acx+aby*acy)*(abx*acx+aby*acy)/(abx*abx+aby*aby)<=r*r)return true;
		}
	}
	return false;
}
bool inclusive(int ax,int ay,int bx,int by,int cx,int cy,int px,int py){
	bool ok1=true;
	bool ok2=true;
	if((bx-ax)*(py-ay)-(px-ax)*(by-ay)<0)ok1=false;
	else ok2=false;
	if((cx-bx)*(py-by)-(px-bx)*(cy-by)<0)ok1=false;
	else ok2=false;
	if((ax-cx)*(py-cy)-(px-cx)*(ay-cy)<0)ok1=false;
	else ok2=false;
	return ok1||ok2;
}
int main(){
	int a,b,c,d,e,f,g,h,r;
	while(scanf("%d%d",&a,&b),a||b){
		scanf("%d%d%d%d%d%d%d",&c,&d,&e,&f,&g,&h,&r);
		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)printf("b\n");
		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))printf("c\n");
		else if(inclusive(a,b,c,d,e,f,g,h))printf("a\n");
		else printf("d\n");
	}
}