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.FileInputStream;
import java.io.FileNotFoundException;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.InputMismatchException;

public class Main {
	InputStream is;

	boolean __FILE_DEBUG_FLAG__ = false;
	String __DEBUG_FILE_NAME__ = "src/2156-input.txt";

	FastScanner in;
	PrintWriter out;
	
	private final int SORT_TYPE_X = 1;
	private final int SORT_TYPE_Y = 2;
	
	private int points_sort_type = -1;
	private double EPS = 1e-9;
	
	double dadd(double a, double b) {
		if (Math.abs(a + b) < EPS * (Math.abs(a) + Math.abs(b))) return 0;
		return a + b;
	}
	
	class SortTypeUndefinedException extends Exception {
		
	}
	
	class Point implements Comparable<Point> {
		double x;
		double y;
		
 		Point(double x, double y) {
 			this.x = x;
 			this.y = y;
		}
		
		Point add(double sx, double sy) {
			return new Point(dadd(x, sx), dadd(y, sy));
		}
		
		Point add(Point p) {
			return add(p.x, p.y);
		}
		
		Point sub(double sx, double sy) {
			return new Point(dadd(x, -sx), dadd(y, -sy));			
		}
		
		Point sub(Point p) {
			return sub(p.x, p.y);
		}
		
		Point mult(double k) {
			return new Point(x * k, y * k);
		}
		
		double dot(Point p) {
			return dadd(x * p.x, y * p.y);
		}

		double det(Point p) {
			return dadd(x * p.y, y * p.x);
		}
				
		double sin(Point p) {
			double cosVal = cos(p);
			return Math.sqrt(1 - cosVal * cosVal);
		}
		
		double cos(Point p) {
			double res = dot(p) / (abs() * p.abs());
			if (res > 1) return res - EPS;
			if (res < -1) return res + EPS;
			return res;
		}
		
		double abs() {
			return Math.sqrt(dadd(x * x, y * y));
		}
		
		public int compareTo(Point s) {
			try {
				switch (points_sort_type) {
				case SORT_TYPE_X :
					if (x == s.x) return Double.compare(y, s.y);
					return Double.compare(x, s.x);
				case SORT_TYPE_Y :
					if (y == s.y) return Double.compare(x, s.x);
					return Double.compare(y, s.y);
				default :
					throw new SortTypeUndefinedException();
				}
			} catch (SortTypeUndefinedException e) {
				e.printStackTrace();
				System.exit(1);
			}
			
			return -1;
		}

		public String toString() {
			return "(" + x + ", " + y + ")]";
		}
	}
	
	Polygon getConvexHull(Point[] ps) {
		points_sort_type = SORT_TYPE_Y;
		Arrays.sort(ps);
		int k = 0, n = ps.length;
		Point[] qs = new Point[n * 2];
		
		for (int i = 0; i < n; i++) {
			while (k > 1 && qs[k-1].sub(qs[k-2]).det(ps[i].sub(qs[k-1])) <= 0) k--;
			
			qs[k++] = ps[i];
		}
		
		for (int i = n - 2, t = k; i >= 0; i--) {
			while (k > t && qs[k-1].sub(qs[k-2]).det(ps[i].sub(qs[k-1])) <= 0) k--;
			
			qs[k++] = ps[i];
		}
		qs = Arrays.copyOf(qs, k - 1);
		
		return new Polygon(qs);
	}
	
	class Polygon {
		Point[] ps;
		
		Polygon(Point[] ps) {
			this.ps = ps;
		}
		
		int getSize() {
			return ps.length;
		}
		
		double getArea() {
			int n = getSize();
			double res = 0;
			for (int i = 0; i < n; i++) {
				res += (ps[(i + 1) % n].y + ps[i].y) * (ps[i].x - ps[(i + 1) % n].x) / 2.0;    
		    }
			return Math.abs(res);
		}
		
		double getDistance() {
			int n = getSize();
			double res = 0;
			for (int i = 0; i < n; i++) {
				res += ps[i].sub(ps[(i+1)%n]).abs();
			}
			return res;
		}

		Point calcCenterOfGravity() {
			Point res = new Point(0, 0);
			int n = getSize();
			
			for (int i = 0; i < n; i++) {
				res = res.add(ps[i]);
			}
			
			res = res.mult(1.0 / n);
			return res;
		}
		
		public String toString() {
			StringBuilder res = new StringBuilder();
			res.append("Polygon : " + ps.length + " <");;
			for (Point p : ps) {
				res.append(p.toString() + ", ");
			}
			res.append(">\n");
			return res.toString();
		}
	}
	
	public void solve() {	
		int n = in.nextInt();
		if (n % 2 == 1) {
			System.out.println("NA");
			return;
		}
		Point[] ps = new Point[n];
		for (int i = 0; i < n; i++) {
			ps[i] = new Point(in.nextDouble(), in.nextDouble());
		}
		Polygon pol = new Polygon(ps);

		double cx = ps[0].x + ps[n/2].x;
		double cy = ps[0].y + ps[n/2].y;
		for (int i = 1; i < n/2; i++) {
			double cx2 = ps[i].x + ps[n/2+i].x;
			double cy2 = ps[i].y + ps[n/2+i].y;
			if (!(cx == cx2 && cy == cy2)) {
				System.out.println("NA");
				return;
			}
		}
		
		Point pg = pol.calcCenterOfGravity();
		System.out.println(pg.x + " " + pg.y);
	}

	
	public void run() {
		if (__FILE_DEBUG_FLAG__) {
			try {
				is = new FileInputStream(__DEBUG_FILE_NAME__);
			} catch (FileNotFoundException e) {
				e.printStackTrace();
			}
			System.out.println("FILE_INPUT!!");
		} else {
			is = System.in;
		}
		in = new FastScanner(is);
		out = new PrintWriter(System.out);

		solve();
	}

	public static void main(String[] args) {
		new Main().run();
	}

	public void mapDebug(int[][] a) {
		System.out.println("--------map display---------");

		for (int i = 0; i < a.length; i++) {
			for (int j = 0; j < a[i].length; j++) {
				System.out.printf("%3d ", a[i][j]);
			}
			System.out.println();
		}

		System.out.println("----------------------------");
		System.out.println();
	}

	public void debug(Object... obj) {
		System.out.println(Arrays.deepToString(obj));
	}

	class FastScanner {
		private InputStream stream;
		private byte[] buf = new byte[1024];
		private int curChar;
		private int numChars;

		public FastScanner(InputStream stream) {
			this.stream = stream;
			//stream = new FileInputStream(new File("dec.in"));

		}

		int read() {
			if (numChars == -1)
				throw new InputMismatchException();
			if (curChar >= numChars) {
				curChar = 0;
				try {
					numChars = stream.read(buf);
				} catch (IOException e) {
					throw new InputMismatchException();
				}
				if (numChars <= 0)
					return -1;
			}
			return buf[curChar++];
		}

		boolean isSpaceChar(int c) {
			return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
		}

		boolean isEndline(int c) {
			return c == '\n' || c == '\r' || c == -1;
		}

		int nextInt() {
			return Integer.parseInt(next());
		}

		int[] nextIntArray(int n) {
			int[] array = new int[n];
			for (int i = 0; i < n; i++)
				array[i] = nextInt();

			return array;
		}

		int[][] nextIntMap(int n, int m) {
			int[][] map = new int[n][m];
			for (int i = 0; i < n; i++) {
				map[i] = in.nextIntArray(m);
			}
			return map;
		}

		long nextLong() {
			return Long.parseLong(next());
		}

		long[] nextLongArray(int n) {
			long[] array = new long[n];
			for (int i = 0; i < n; i++)
				array[i] = nextLong();

			return array;
		}

		long[][] nextLongMap(int n, int m) {
			long[][] map = new long[n][m];
			for (int i = 0; i < n; i++) {
				map[i] = in.nextLongArray(m);
			}
			return map;
		}

		double nextDouble() {
			return Double.parseDouble(next());
		}

		double[] nextDoubleArray(int n) {
			double[] array = new double[n];
			for (int i = 0; i < n; i++)
				array[i] = nextDouble();

			return array;
		}

		double[][] nextDoubleMap(int n, int m) {
			double[][] map = new double[n][m];
			for (int i = 0; i < n; i++) {
				map[i] = in.nextDoubleArray(m);
			}
			return map;
		}

		String next() {
			int c = read();
			while (isSpaceChar(c))
				c = read();
			StringBuilder res = new StringBuilder();
			do {
				res.appendCodePoint(c);
				c = read();
			} while (!isSpaceChar(c));
			return res.toString();
		}

		String[] nextStringArray(int n) {
			String[] array = new String[n];
			for (int i = 0; i < n; i++)
				array[i] = next();

			return array;
		}

		String nextLine() {
			int c = read();
			while (isEndline(c))
				c = read();
			StringBuilder res = new StringBuilder();
			do {
				res.appendCodePoint(c);
				c = read();
			} while (!isEndline(c));
			return res.toString();
		}
	}
}