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 <cstring>
#include <algorithm>
#include <string>
#include <cmath>
#include <stack>
#include <vector>
#include <map>
#include <set>
#include <queue>
#include <list>
#include <iostream>
#include <sstream>
#include <climits>
#include <cfloat>
#include <complex>


typedef long long ll;
const double Eps = 1e-4;

using namespace std;

typedef complex<double> Point;

double dot(const Point& a, const Point& b)
{
	return a.real()*b.real() + a.imag()*b.imag();
}
double cross(const Point& a, const Point& b)
{
	return a.real()*b.imag() - a.imag()*b.real();
}
double distance_line_point(const Point& point, const Point& a, const Point& b)
{
	return abs(cross(a-b, point-b) / abs(a-b));
}
double distance_linesegment_point(const Point& point, const Point& a, const Point& b)
{
	if (dot(b-a, point-a) < Eps)
		return abs(a-point);
	else if (dot(a-b, point-b) <Eps)
		return abs(b-point);
	else
		return distance_line_point(a, b, point);
}
bool is_in_triangle(const Point& point, const Point* triangle)
{
	double t[3];
	for (int i = 0; i < 3; ++i)
		t[i] = cross(triangle[(i+1)%3] - triangle[i], point - triangle[i]);

	return t[0] >= 0 && t[1] >= 0 && t[2] >= 0
		|| t[0] <= 0 && t[1] <= 0 && t[2] <= 0;
}
bool is_in_triangle(const Point& point, const Point& a, const Point& b, const Point& c)
{
	Point t[3] = { a, b, c };
	return is_in_triangle(point, t);
}
bool is_circle_in_triangle(const Point& center, double radius, const Point* triangle)
{
	if (!is_in_triangle(center, triangle))
		return false;
	for (int i = 0; i < 3; ++i)
		if (distance_line_point(triangle[i], triangle[(i+1)%3], center) < radius)
			return false;
	return true;
}
bool is_triangle_in_circle(const Point* triangle, const Point& center, double radius)
{
	for (int i = 0; i < 3; ++i)
		if (abs(triangle[i]-center) > radius)
			return false;
	return true;
}
bool interect_triangle_circle(const Point* triangle, const Point& center, double radius)
{
	for (int i = 0; i < 3; ++i)
		if (distance_linesegment_point(center, triangle[i], triangle[(i+1)%3]) < radius)
			return true;
	return false;
}

int main()
{
	int x[4], y[4], r;
	while (true)
	{
		for (int i = 0; i < 4; ++i)
			scanf("%d %d", x+i, y+i);
		scanf("%d", &r);
		if (x[0] == 0 && y[0] == 0)
			break;
		Point p[4];
		for (int i = 0; i < 4; ++i)
			p[i] = Point(x[i], y[i]);

		if (is_circle_in_triangle(p[3], r, p))
			puts("a");
		else if (is_triangle_in_circle(p, p[3], r))
			puts("b");
		else if (interect_triangle_circle(p, p[3], r))
			puts("c");
		else
			puts("d");
	}

	return 0;
}

C++: #include <iostream>
#include <cstdio>
#include <cmath>
#include <algorithm>
using namespace std;

double D(double x, double y){
	return sqrt(x * x + y * y);
}

double inp(double x1, double x2, double y1, double y2){
	return x1 * x2 + y1 * y2;
}

int main(){
	int x1, y1, x2, y2, x3, y3, x4, y4, r;
	while( cin >> x1 >> y1 , x1 || y1 ){
		cin >> x2 >> y2 >> x3 >> y3 >> x4 >> y4 >> r;
		x1 -= x4; x2 -= x4; x3 -= x4;
		y1 -= y4; y2 -= y4; y3 -= y4;
		
		double d = max(D(x1, y1), max(D(x2, y2), D(x3, y3)));
		if( d <= r ){
			puts("b");
		}else{
			double d1, d2, d3;
			if(D(x1, y1) < inp(x1, x2, y1, y2) || D(x2, y2) < inp(x1, x2, y1, y2) ){
				d1 = D(x1, y1);
			}else{
				d1 = fabs(x1 * y2 - y1 * x2) / D(x2 - x1, y2 - y1);
			}
			if(D(x2, y2) < inp(x2, x3, y2, y3) || D(x3, y3) < inp(x2, x3, y2, y3) ){
				d2 = D(x2, y2);
			}else{
				d2 = fabs(x2 * y3 - y2 * x3) / D(x3 - x2, y3 - y2);
			}
			if(D(x3, y3) < inp(x3, x1, y3, y1) || D(x1, y1) < inp(x3, x1, y3, y1) ){
				d3 = D(x3, y3);
			}else{
				d3 = fabs(x3 * y1 - y3 * x1) / D(x1 - x3, y1 - y3);
			}
			double d = min(d1, min(d2, d3));
			int t = (x1 * y2 > y1 * x2) + (x2 * y3 > y2 * x3) + (x3 * y1 > y3 * x1);
			bool in = !(t && (t-3));
			
			if( in ){
				if( d < r ){
					puts("c");
				}else{
					puts("a");
				}
			}else{
				if( d <= r ){
					puts("c");
				}else{
					puts("d");
				}
			}
		} 
	}
}