Interface Shape and class Triangle : Shape

#include <iostream>
#include <cmath>
#include <vector>
#include <exception>

using namespace std;

struct Shape {
    virtual double area() = 0;
    virtual double perimeter() = 0;
    virtual string toString() = 0;
    string Type() {
        return type;
    }
protected:
    string type;
public:
    struct Point {
        string toString() {
            string result = "{";
            for (int i = 0; i < getDimension(); i ++) {
                result += to_string(coordinates[i]);
                if (i < getDimension() - 1) {
                    result += ", ";
                }
            }
            result += "}";
            return result;
        }

        vector<double> coordinates;

        Point(vector<double> input) : coordinates (input) {}

        unsigned int getDimension() {
            return coordinates.size();
        }

        double distance() {
            double sum = 0;
            for (auto x = coordinates.begin(); x != coordinates.end(); x ++) {
                sum += pow(*x, 2);
            }
            return sqrt(sum);
        }

        Point& operator- (Point B) {
            if (B.getDimension() != getDimension()) {
                throw new exception();
            }
            vector<double> newCoordinates(getDimension());
            for (int i = 0; i < getDimension(); i ++) {
                newCoordinates[i] = coordinates[i] - B.coordinates[i];
            }
            Point* result = new Point{newCoordinates};
            return *result;
        }

        Point& operator+ (Point B) {
            if (B.getDimension() != getDimension()) {
                throw new exception();
            }
            vector<double> newCoordinates(getDimension());
            for (int i = 0; i < getDimension(); i ++) {
                newCoordinates[i] = coordinates[i] + B.coordinates[i];
            }
            Point* result = new Point(newCoordinates);
            return *result;
        }

        Point& operator/ (double x) {
            for (int i = 0; i < getDimension(); i ++) {
                coordinates[i] /= x;
            }
            return *this;
        }

        double distance (Point B) {
            return (*this - B).distance();
        }
    };
    virtual Shape::Point center() = 0;
};

struct Triangle : Shape {
    string toString() {
        string result = "";
        result += "\nA: ";
        result += A.toString();
        result += "\nB: ";
        result += B.toString();
        result += "\nC: ";
        result += C.toString();
        return result;
    }
    Triangle(vector< vector <double> > coordinates) : A(coordinates[0]), B(coordinates[1]), C(coordinates[2]) {
        if (A.getDimension() != B.getDimension() || A.getDimension() != C.getDimension()) {
            throw new exception();
        }
        type = "Triangle";
    }
    double perimeter() {
        return A.distance(B) + B.distance(C) + C.distance(A);
    }
    double area() {
        double P = perimeter()/2;
        return sqrt(P * (P - A.distance(B)) * (P - B.distance(C)) * (P - C.distance(A)));
    }
    Point center() {
        return (A + B + C)/3;
    }
private:
    Point A;
    Point B;
    Point C;
};

int main()
{
    vector< vector<double> > coordinates(3);
    coordinates[0] = {0, 0};
    coordinates[1] = {0, 5};
    coordinates[2] = {12, 0};
    Triangle ABC(coordinates);
    cout << ABC.Type() << ABC.toString();
    cout << "\nArea: " << ABC.area();
    cout << "\nPerimeter: " << ABC.perimeter();
    cout << "\nCenter: " << ABC.center().toString();
    return 0;
}