Centroid of Point Masses

The centroid of a region in the plane can be thought of as the point where the region would balance on the tip of a pencil. Working that out for an arbitrary region takes far more effort than this problem calls for, so we only look for the centroid of a collection of point masses.

Here the plane is a thin sheet with no mass of its own, carrying a few heavy points, and the question is where the sheet would balance. The precise definition follows.

Given the coordinates $(x_i, y_i)$ of $n$ points and the mass $m_i$ at each point, the x-moment of that set relative to a point $(a, b)$ is

$M_x = \sum_{i=1}^{n} m_i (b - y_i)$

Note that the x-moment is defined with differences of $y$ coordinates. This problem uses both moments, so it does not matter which name goes on which sum.

The y-moment is defined the same way.

$M_y = \sum_{i=1}^{n} m_i (a - x_i)$

The centroid of the set is the point $(a, b)$ at which both moments are zero.

The input holds several sets of points. Each set starts with the number of points $n$, followed by $n$ lines giving $x_i$, $y_i$ and $m_i$ for each point. A negative value of $n$ ends the input.

• $1 \le n \le 5000$
• $0 \le x_i, y_i \le 5000$
• $1 \le m_i \le 5000$

All values are integers. The input carries extra whitespace so that a person can read it easily, so do not assume any particular number of spaces before, between or after the values, and do not assume a particular number of blank lines between sets.

For each set, print the coordinates of the centroid on one line. Follow the format exactly: "Case", one space, the set number starting at 1, a colon and one space, then $a$ and $b$ rounded to two decimal places and separated by one space. Rounding happens at the third decimal place, and an exact half rounds up. No input puts an exact value on a rounding boundary. Do not print trailing spaces.