Brocard Point of a Triangle

The Brocard point of a triangle $ABC$ is the point $P$ inside the triangle with $\angle PAB = \angle PBC = \angle PCA$. See the figure below.

The size the three angles share is called the Brocard angle. The Brocard angle is at most $\pi/6$, and that maximum occurs for an equilateral triangle. The Brocard point of an equilateral triangle is its centroid.

Given the coordinates of the three vertices, write a program that computes the coordinates of the Brocard point.

The first line contains the number of data sets $P$ ($1 \le P \le 10000$). The data sets are independent and every one is processed the same way.

Each data set is a single line. The line begins with the data set number $K$, followed by the coordinates $A_x$, $A_y$, $B_x$, $B_y$, $C_x$, $C_y$ of the three vertices, separated by spaces. $K$ is an integer between $1$ and $P$. Each coordinate is a real number whose absolute value is at most $1000$.

The vertices are always given so that going from $A$ to $B$, from $B$ to $C$, and from $C$ back to $A$ circles the triangle counter-clockwise. The three points are never collinear.

Print one line for each data set. Print the data set number $K$, a space, the $x$ coordinate of the Brocard point, a space, and the $y$ coordinate of the Brocard point.

Round both coordinates to five decimal places. If a rounded coordinate is zero, print 0.00000 and never -0.00000.