Problem:
Convex-Cut

N個の頂点からなる凸多角形が与えられる。各頂点の座標は反時計周りに(X1, Y1), (X2 ,Y2), ……, (XN, YN)で表わされる。
点Pを通るどのような直線で凸多角形を切断しても、切断後に得られる2つの凸多角形の面積が等しいような点Pの座標を求めよ。

Input

入力は以下の形式で与えられる。

N
X1 Y1
X2 Y2
……
XN YN

Constraints

入力は全て整数である

3 ≤ N ≤ 50

0 ≤ |Xi|, |Yi| ≤ 1000000

入力の多角形は単純な凸多角形である。

出力は、出力座標を(X, Y)、厳密解を(cX, cY)とした時にmax(|X-cX|, |Y-cY|) ≤ 0.0001を満たす必要がある

Output

問題文の条件を満たす点があるのならば、その点の座標を

X Y

の形式で出力せよ。
点が存在しない場合は"NA"と1行に出力せよ。

Sample Input 1

4
100 100
0 100
0 0
100 0

以下の図に対応する。

Output for the Sample Input 1

50.00000 50.00000

Sample Input 2

3
100 100
0 100
0 0

Output for the Sample Input 2

NA

以下の図に対応する。

Solution in JAVA (judge verdict: Accepted)

import java.io.IOException;
import java.util.LinkedList;
import java.util.List;
import java.util.Scanner;
 
 
public class Main{
	
    public static void main(String[] args) throws IOException {
    	Scanner sc = new Scanner(System.in);
    	
    	final int N = sc.nextInt();
    	
    	Point2D[] points = new Point2D[N];
    	for(int i = 0; i < N; i++){
    		points[i] = new Point2D(sc.nextInt(), sc.nextInt());
    	}
    	
    	if(N % 2 != 0){
    		System.out.println("NA");
    		return;
    	}
    	
    	Point2D first = Point2D.interpoint_s(points[0], points[0 + N/2], points[1], points[1 + N/2]);
    	
    	for(int i = 2; i < N/2; i++){
    		if(!first.equals(Point2D.interpoint_s(points[0], points[0 + N/2], points[i], points[i + N/2]))){
    			System.out.println("NA");
    			return;
    		}
    	}
    	
    	System.out.println(first.x + " " + first.y);
    }
     
}

class Point2D {
	public double x;
	public double y;
	
	public static final double EPS = 1e-6;
	
	public Point2D(double x, double y){
		this.x = x;
		this.y = y;
	}
	
	public Point2D(Point2D point){
		this.x = point.x;
		this.y = point.y;
	}
	
	public String toString(){
		return x + "," + y;
	}
	
	@Override
	public boolean equals(Object o){
		if(o instanceof Point2D){
			Point2D another = (Point2D) o;
			
			if(this.x - EPS < another.x && this.x + EPS > another.x &&
				this.y - EPS < another.y && this.y + EPS > another.y){
				return true;
			}
			
			return false;
			//return this.x == another.x && this.y == another.y;
		}
		return false;
	}
	
	public Point2D add(double x, double y){
		return new Point2D(this.x + x, this.y + y);
	}
	
	public Point2D sub(double x, double y){
		return add(-x, -y);
	}
	
	public Point2D add(Point2D another){
		return add(another.x, another.y);
	}
	
	public Point2D sub(Point2D another){
		return sub(another.x, another.y);
	}
	
	public Point2D mul(double d){
		return new Point2D(this.x * d, this.y * d);
	}
	
	public Point2D div(double d){
		return new Point2D(this.x / d, this.y / d);
	}
	
	public double dot(double x, double y){
		return this.x * x + this.y * y; 
	}
	
	public double dot(Point2D another){
		return dot(another.x, another.y);
	}
	
	public double cross(double x, double y){
		return this.x * y - this.y * x; 
	}
	
	public double cross(Point2D another){
		return cross(another.x, another.y);
	}
	
	public double dist(double x, double y){
		return Math.sqrt((this.x - x)*(this.x - x) + (this.y - y) * (this.y - y));
	}
	
	public double dist(Point2D another){
		return dist(another.x, another.y);
	}
	
	public double dist_o(){
		return dist(0, 0);
	}
	
	public Point2D unit(){
		return div(dist_o());
	}
	
	public boolean pol(Point2D start, Point2D end){
		return end.sub(start).cross(this.sub(start)) < EPS;
	}
	
	public boolean pos(Point2D start, Point2D end){
		return (start.dist(this) + this.dist(end) < start.dist(end) + EPS);
	}
	
	public double pld(Point2D start, Point2D end){
		return Math.abs((end.sub(start).cross(this.sub(start)))/end.sub(start).dist_o());
	}
	
	public double psd(Point2D start, Point2D end){
		if(end.sub(start).dot(this.sub(start)) < EPS){
			return this.dist(start);
		}else if(start.sub(end).dot(this.sub(end)) < EPS){
			return this.dist(end);
		}else{
			return end.sub(start).cross(this.sub(start)) / end.dist(start);
		}
	}
	
	public static boolean intersect_s(Point2D a1, Point2D a2, Point2D b1, Point2D b2){
		return (a2.sub(a1).cross(b1.sub(a1)) * a2.sub(a1).cross(b2.sub(a1)) < EPS) && (b2.sub(b1).cross(a1.sub(b1)) * b2.sub(b1).cross(a2.sub(b1)) < EPS);
	}
	
	public static boolean insersect_l(Point2D a1, Point2D a2, Point2D b1, Point2D b2){
		return a1.sub(a2).cross(b1.sub(b2)) < EPS;
	}
	
	public static Point2D interpoint_s(Point2D a1, Point2D a2, Point2D b1, Point2D b2){
		Point2D b = b2.sub(b1);
		double d1 = Math.abs(b.cross(a1.sub(b1)));
		double d2 = Math.abs(b.cross(a2.sub(b1)));
		double t = d1 / (d1 + d2);
		Point2D a = a2.sub(a1), v = a.mul(t);
		return a1.add(v);
	}
	
	public static Point2D interpoint_l(Point2D a1, Point2D a2, Point2D b1, Point2D b2){
		Point2D a = a2.sub(a1);
		Point2D b = b2.sub(b1);
		double t = b.cross(b1.sub(a1)) / b.cross(a);
		Point2D v = a.mul(t);
		return a1.add(v);
	}
	
	public static Point2D[] cross_ss(Point2D p1, double r1, Point2D p2, double r2){
		double dis = p1.dist(p2);
		
		if(r1 + EPS > r2 && r1 - EPS < r2 && dis < EPS){
			return new Point2D[0]; //same
		}
		
		
		if(dis - EPS < r1 + r2 && dis + EPS > r1 + r2){
			Point2D tmp = p2.sub(p1);
			tmp = tmp.mul(r1 / tmp.dist_o());
			Point2D ret[] = new Point2D[1];
			ret[0] = p1.add(tmp);
			return ret;
		}else if(dis + EPS > r1 + r2){
			return new Point2D[0]; //out
		}
		
		double dis_m = Math.abs(r1-r2);
		
		
		if(dis_m + EPS > dis && dis_m - EPS < dis){
			Point2D tmp = null;
			if(r1 > r2){
				tmp = p2.sub(p1);
			}else{
				tmp = p1.sub(p2);
			}
			
			double min = Math.min(r1, r2);
			
			tmp = tmp.mul((min + tmp.dist_o()) / tmp.dist_o());
			
			Point2D ret[] = new Point2D[1];
			ret[0] = p1.add(tmp);
			return ret;
		}else if(dis_m + EPS > dis){
			return new Point2D[0]; //inner
		}else{
			Point2D ret[] = new Point2D[2];
			
			double theta = Math.acos((dis*dis + r1*r1 - r2*r2)/(2*dis*r1));
			double a = Math.atan2(p2.y-p1.y, p2.x-p1.x);
			
			ret[0] = new Point2D(r1*Math.cos(a+theta) + p1.x, r1*Math.sin(a+theta) + p1.y);
			ret[1] = new Point2D(r1*Math.cos(a-theta) + p1.x, r1*Math.sin(a-theta) + p1.y);
			return ret;
		}	
	}
	
	public void interpoint_lc(Point2D start , Point2D end , Point2D c , double r , Point2D ans[]){
		if(c.pld(start , end) > r + EPS) return;
		Point2D v= end.sub(start).unit();
		double delta=v.dot(start.sub(c))*v.dot(start.sub(c))-start.dist(c)*start.dist(c)+r*r;
		double t=-v.dot(start.sub(c));
		double s= Math.sqrt(delta);
		ans[0]=start.add(v.mul(t+s));
		ans[1]=start.add(v.mul(t+s));
	}
	
	public Point2D normal_vector(Point2D p,Point2D a,Point2D b){
		Point2D v= b.sub(a).unit();
		v = v.cross(p.sub(a)) > 0 ?  new Point2D(v.y,(-1)*v.x) : new Point2D((-1)*v.y , v.x);
		return v.mul(p.pld(a,b));
	}
	
	public double area(Point2D a,Point2D b,Point2D c){
		return Math.abs((c.sub(a).cross(b.sub(a)))*0.5);
	}
	
}