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 <algorithm>
#include <complex>
#include <vector>
#include <cmath>
 
using namespace std;
 
#define F first
#define S second
 
#define X real()
#define Y imag()
 
#define EPS (1e-7)
 
typedef complex<double> Point;
typedef vector<Point> Polygon;
typedef pair<double, Point> Circle;
 
bool isPointInCircle(const Point& p, const Circle& c) {
  if(abs(p-c.S) < abs(c.F) + EPS) return true;
  return false;
}
 
double dot(const Point& a, const Point& b) {
  return a.X*b.X+a.Y*b.Y;
}
 
double cross(const Point& a, const Point& b) {
  return a.X*b.Y-a.Y*b.X;
}
 
typedef pair<Point, Point> Segment;
typedef Segment Line;
 
double distanceLP(Line l, Point p) {
  return abs(cross(l.S-l.F, p-l.F)) / abs(l.S-l.F);
}
 
double distanceSP(Segment s, Point p) {
  Point a = s.F, b = s.S;
   
  if(dot(b-a, p-a) < EPS) return abs(p-a);
  if(dot(a-b, p-b) < EPS) return abs(p-b);
  return distanceLP(s, p);
}
 
 
enum struct ECont { OUT, ON, IN };
ECont contains(const Polygon& poly, const Point& p) {
  bool in = 0;
  for(int i=0; i<poly.size(); i++) {
    Point a = poly[i] - p, b = poly[(i+1)%poly.size()] - p;
    if(a.imag() > b.imag()) swap(a, b);
    if(a.imag() <= 0 && 0 < b.imag()) {
      if(cross(a, b) < 0) in = !in;
    }
    if(cross(a, b) == 0 && dot(a, b) <= 0) return ECont::ON;
  }
   
  return in ? ECont::IN : ECont::OUT;
}
 
int main() {
   
  while(1) {
    Polygon points(3);
    Circle circle;
   
    for(int i=0; i<3; i++) {
      double x, y; cin >> x >> y;
      if(x == 0) return 0;
     
      points[i] = Point(x, y);
    }
   
    {
      double x, y; cin >> x >> y;
      circle.S = Point(x, y);
      cin >> circle.F;
    }
   
    bool ok = 1;
    for(int i=0; i<3; i++) {
      ok &= isPointInCircle(points[i], circle);
    }
    if(ok) {
      cout << 'b' << endl;
      continue;
    }
     
    int uncross = 0;
    for(int i=0; i<3; i++) {
      Segment seg = make_pair(points[(i+1)%3], points[i]);
      uncross += circle.F < distanceSP(seg, circle.S) + EPS;
    }
    if(uncross == 3 && contains(points, circle.S) == ECont::IN) {cout << 'a' << endl; continue;}
 
    int cross = 0;
    for(int i=0; i<3; i++) {
      Segment seg = make_pair(points[(i+1)%3], points[i]);
      cross += distanceSP(seg, circle.S) <= circle.F + EPS;
    }
    if(cross > 0) cout << 'c' << endl;
    else cout << 'd' << endl;
  }
   
  return 0;
}

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

#define EPS 1e-6
typedef complex<double> P;

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

struct C {
  P p; double r;
  C(const P &p, double r) : p(p), r(r) { }
};

double cross(const P& a, const P& b) {
  return imag(conj(a)*b);
}

double dot(const P& a, const P& b) {
  return real(conj(a)*b);
}

double dist(const P& a, const P& b)
{
	return sqrt((a.real()-b.real())*(a.real()-b.real())+(a.imag()-b.imag())*(a.imag()-b.imag()));
}

bool isIntoC(const P& p, const C& c)
{
	return dist(p, c.p) < c.r+EPS;
}

bool isIntoC(const L& l, const C& c)
{
	return dist(l[0], c.p) < c.r+EPS && dist(l[1], c.p) < c.r+EPS;
}

int CrossPtNum(const C& c, const L& l)
{
	P LV=l[0]-l[1];
	double A=(LV.real()*LV.real())+(LV.imag()*LV.imag());
	double B=(LV.real()*l[0].real())-(LV.real()*c.p.real())+(LV.imag()*l[0].imag())-(LV.imag()*c.p.imag());
	double C=(l[0].real()-c.p.real())*(l[0].real()-c.p.real())+(l[0].imag()-c.p.imag())*(l[0].imag()-c.p.imag())-(c.r*c.r);

	if(fabs(B*B-A*C) < EPS) return 1;
	if(B*B-A*C  > 0) return 2;
	
	return 0;
}

bool intersectSC(const L& l, const C& c)
{
	if(isIntoC(l, c)) return false;
	
	double vc=dot(l[1]-l[0], c.p-l[0]);
	if(vc<0)
	{
		double dif=c.r-dist(l[0], c.p);
		return dif+EPS > 0;
	}
	
	double vv=dot(l[1]-l[0], l[1]-l[0]);
	
	if(vc > vv)
	{
		double d=dist(l[1], c.p);
		double dif=c.r*c.r-d*d;
		
		return dif+EPS > 0;
	}
	
	double cc=dot(c.p-l[0], c.p-l[0]);
	double dif=c.r*c.r-(cc-(vc/vv)*vc);
	
	return dif+EPS > 0;
}

int main()
{
	int x,y,r;
	while(scanf("%d%d", &x, &y), (x||y))
	{
		vector<P> tri;
		for(int i=0; i<3; i++)
		{
			tri.push_back(P(x,y));
			if(i==2) break;
			scanf("%d%d", &x, &y);
		}
	
		scanf("%d%d%d", &x,&y,&r);
		C c(P(x,y), r);
		
		bool g=true;
		for(int i=0; i<3; i++)
		{
			if(!isIntoC(tri[i], c))
			{
				g=false;
				break;
			}
		}
		
		if(g) 
		{
			puts("b");
			continue;
		}
		int cr[3];
		cr[0]=cross(tri[0]-tri[1], tri[0]-c.p);
		cr[1]=cross(tri[1]-tri[2], tri[1]-c.p);
		cr[2]=cross(tri[2]-tri[0], tri[2]-c.p);
		
		g=true;
		if(cr[0]<EPS&&cr[1]<EPS&&cr[2]<EPS||cr[0]+EPS>0&&cr[1]+EPS>0&&cr[2]+EPS>0)
		{
			for(int i=0; i<3; i++)
			{
				if(CrossPtNum(c, L(tri[i], tri[(i+1)%3])) == 2) g=false;
			}
			if(g) 
			{
				puts("a");
				continue;
			}
			
		}
		
		for(int i=0; i<tri.size(); i++)
		for(int j=i+1; j<tri.size(); j++)
		{
			if(intersectSC(L(tri[i], tri[j]), c))	g=true;
		}
		if(g)
		{
			puts("c");
			continue;
			
		}
		
		puts("d");
	}
}