Triangle to Hexagon

Let $I$ be the incenter of triangle $ABC$ and let $O$ be its circumscribed circle. The lines $AI$, $BI$ and $CI$ meet the circle $O$ a second time at $M$, $N$ and $P$ respectively.

Let $E$ and $F$ be the points where the line $NP$ meets the sides $AB$ and $AC$. In the same way, let $G$ and $H$ be the points where the line $MN$ meets $AC$ and $BC$, and let $J$ and $K$ be the points where the line $MP$ meets $BC$ and $AB$.

Read the coordinates of the vertices $A$, $B$ and $C$, then print the lengths of the segments $EF$, $FG$, $GH$, $HJ$, $JK$ and $KE$. The inequality $|EF| + |GH| + |JK| \le |KE| + |FG| + |HJ|$ is known to hold, so you can use it to check your own numbers.

Do the computation in double precision floating point.

The first line contains one integer $P$, the number of data sets ($1 \le P \le 10000$). Every data set is processed the same way and independently of the others.

Each data set is a single line. The line holds the data set number $K$ followed by three floating point values separated by spaces: the $x$ coordinate of $B$, $B_x$, the $x$ coordinate of $C$, $C_x$, and the $y$ coordinate of $C$, $C_y$. $A$ is always the origin $(0, 0)$ and $B$ always lies on the $x$ axis, so $B_y = 0$.

$K$ is an integer, $0 < B_x \le 1000$, $|C_x| \le 1000$ and $0 < C_y \le 1000$. The three points are never collinear.

Print one line per data set. The line holds the data set number $K$, then the lengths of $EF$, $FG$, $GH$, $HJ$, $JK$ and $KE$ in that order, each rounded to 4 decimal places and separated by a single space.