Brocard Point of a Triangle

The *Brocard* point of a triangle
$ABC$ is a point
$P$ in the triangle chosen
so that: $\angle PAB = \angle PBC
= \angle PCA$ (see figure below).

The common angle is called the *Brocard* angle. The largest *Brocard* angle is $\pi /6$ which is the *Brocard* angle for an equilateral triangle (the
*Brocard* point is the centroid of the
triangle).

Write a program to compute the coordinates of the *Brocard* point of a triangle given the coordinates
of the vertices.

The first line of input contains a single integer $P$, ($1 \le P \le 600$), which is the number of data sets that follow. Each data set should be processed identically and independently.

Each data set consists of a single line of input. It contains the data set number, $K$, followed by the six space separated coordinate values $A_ x, A_ y, B_ x, B_ y, C_ x, C_ y$ of the vertices of the triangle, which are in the range ($-20 \le A_ x, A_ y, B_ x, B_ y, C_ x, C_ y \le 20$). The vertices will always be specified so going from $A$ to $B$ to $C$ and back to $A$ circles the triangle counter-clockwise. Input coordinates are floating point values with up to 4 digits after the period.

For each data set there is a single line of output. The
single output line consists of the data set number,
$K$, followed by a single
space followed by the $x$
coordinate of the *Brocard* point,
followed by a single space followed by the $y$ coordinate of the *Brocard* point. The coordinates are considered
correct if their individual absolute or relative error does not
exceed $10^{-5}$.

Sample Input 1 | Sample Output 1 |
---|---|

3 1 0 -1.3 3.4 0.5 1.1 2.3 2 0 0 3 0 0 4 3 3.1 0.2 4.3 0.4 0 0.8 |
1 1.404561 0.828896 2 1.560468 0.749025 3 3.876994 0.401673 |