#include<iostream>
#include<cmath>
#include<vector>
#include<algorithm>
using namespace std;

enum POSITION {UNDER = -1, ON, ABOVE};

// classes
class Point;
class Segment;
class Circle;

// members
class Point {
public:
    double x, y;
    Point(double x, double y) :x(x), y(y) {}
    const Point& operator=(const Point& p) {x = p.x; y = p.y; return *this;}
    const Point operator+(const Point& p) const {return Point(x + p.x, y + p.y);}
    const Point operator-(const Point& p) const {return Point(x - p.x, y - p.y);}
    const Point operator*(const Point& p) const {return Point(x * p.x, y * p.y);}
    double distance(const Point& p) const;
    double distance(const Segment& s) const;
    double dot(const Point& p) const;
    double cross(const Point& p) const;
    const Point projection(const Segment& s) const;
};
class Segment {
public:
    Point start, end;
    Segment(const Point& s, const Point& e) :start(s), end(e) {}
    POSITION ccw(const Point& p) const {
        const Point& a = start;
        const Point& b = end;
        Point v1 = b - a;
        Point v2 = p - a;
        double d = v1.cross(v2);
        if(d < 0.0) return ABOVE;
        if(d > 0.0) return UNDER;
        return ON;
    }
};
class Circle {
public:
    Point center;
    double r;
    Circle(double x, double y, double r) :center(Point(x, y)), r(r * r) {}
    bool include(const Point& p) const {return center.distance(p) < r;}
    bool include(const Segment& s) const {return center.distance(s) < r;}
};

// implements
double Point::distance(const Point& p) const {return (x - p.x) * (x - p.x) + (y - p.y) * (y - p.y);}
double Point::distance(const Segment& s) const {
    Point p = projection(s);
    if(s.ccw(p) == ON) return distance(p);
    return min(distance(s.start), distance(s.end));
}
double Point::dot(const Point& p) const {return x * p.x - y * p.y;}
double Point::cross(const Point& p) const {return x * p.y - y * p.x;}
const Point Point::projection(const Segment& s) const {
    Point v1 = *this - s.start;
    Point v2 = s.start - s.end;
    double t = v1.dot(v2) / s.start.distance(s.end);
    return s.start + Point(t, t) * v2;
}

// main
string check(const vector<Point>& triangle, const Circle& circle) {
    vector<Segment> edge;
    for(int i = 0; i < 3; ++i) edge.push_back(Segment(triangle[i], triangle[(i + 1) % 3]));

    bool triangleInCircle = true;
    for(auto& p: triangle) triangleInCircle &= circle.include(p);
    if(triangleInCircle) return "b";

    for(auto& e: edge) if(circle.include(e)) return "c";

    int cw = 0, ccw = 0;
    for(auto& e: edge) {
        if(e.ccw(circle.center) >= 0) ++ccw;
        if(e.ccw(circle.center) <= 0) ++cw;
    }
    if(cw == 3 || ccw == 3) return "a";

    return "d";
}

int main() {
    double x[4], y[4], r;
    while(cin >> x[0] >> y[0], x[0] + y[0]) {
        cin >> x[1] >> y[1] >> x[2] >> y[2] >> x[3] >> y[3] >> r;
        vector<Point> triangle = {
            Point(x[0], y[0]),
            Point(x[1], y[1]),
            Point(x[2], y[2]),
        };
        Circle circle(x[3], y[3], r);
        cout << check(triangle, circle) << endl;
    }
}