Lifting the Stone
(file name: stone.c, stone.p, stone.C
)
There are many secret openings in the floor which are covered by a big
heavy stone. When the stone is lifted up, a special mechanism detects this
and activates poisoned arrows that are shot near the opening. The only possibility
is to lift the stone very slowly and carefully. The ACM team must connect
a rope to the stone and then lift it using a pulley. Moreover, the stone
must be lifted all at once; no side can rise before another. So it is very
important to find the centre of gravity and connect the rope exactly to that
point. The stone has a polygonal shape and its height is the same throughout
the whole polygonal area. Your task is to find the centre of gravity for
the given polygon.
Input Specification
The input consists of
T test cases. The number of them (
T) is given on the first line of the input file. Each test case begins
with a line containing a single integer
N (
3 <= N <=
1000000) indicating the number of points that form the polygon. This
is followed by
N lines, each containing two integers
X
i and
Yi (
|Xi|, |Y
i| <= 20000). These numbers are the coordinates of the
i-th point. When we connect the points in the given order, we get
a polygon. You may assume that the edges never touch each other (except the
neighbouring ones) and that they never cross. The area of the polygon is never
zero, i.e. it cannot collapse into a single line.
Output Specification
Print exactly one line for each test case. The line should contain exactly
two numbers separated by one space. These numbers are the coordinates of
the centre of gravity. Round the coordinates to the nearest number with exactly
two digits after the decimal point (0.005 rounds up to 0.01). Note that
the centre of gravity may be outside the polygon, if its shape is not convex.
If there is such a case in the input data, print the centre anyway.
Sample Input
2
4
5 0
0 5
-5 0
0 -5
4
1 1
11 1
11 11
1 11
Output for the Sample Input
0.00 0.00
6.00 6.00