J. Jumpin' Jack Flash
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output

An evil race of aliens called the Rhombulans have invaded planet Earth! Rhombulans are characterized by two main traits. They hate music, to the point where their mission is to exterminate all music in all its forms throughout the galaxy. On the other hand, they love geometry! Which shape do they love the most? Well isn't it obvious...? It's the parallelogram!

The Rhombulans worship the parallelogram, to the point where it has bled into their war strategies, even at times where it would not be the most rational decision. Alice, Bob, and Cindy have been contracted as Elite Bit Agents, and it is their job to sort through all the binary code from the computers of defeated Rhombulans in order to gleam some useful military secrets.

Alice, Bob, and Cindy have unearthed that Rhombulan bases are located at the three non-collinear integer points $$$(x_1, y_1)$$$, $$$(x_2, y_2)$$$, and $$$(x_3, y_3)$$$. A completing point is one such that if they take it together with the three input points, then the four points form the vertices of a parallelogram; Alice, Bob, and Cindy believe that the final Rhombulan base is located at one such completing point.

Now, they need to give a report on the information that they have discovered regarding the possible locations of the final Rhombulan base. After all, there may be multiple possible completing points, and their generals need as much information as they can get in order to best decide where to send their troops.

Thus, you were assigned the following task. For each possible completing point, output the following pieces of information:

  • What are the coordinates of this completing point?
  • What is the area of the formed parallelogram?
  • Is the formed parallelogram a rhombus?
  • Is the formed parallelogram a rectangle?

Godspeed, agent. And remember: music LIVES!

Input

Input consists of three lines, each containing the space-separated integers $$$x_1$$$ and $$$y_1$$$, then $$$x_2$$$ and $$$y_2$$$, then $$$x_3$$$ and $$$y_3$$$.

$$$$$$\begin{align*}

&\begin{array}{|l|} \hline \text{Constraints For All Subtasks} \\ \hline -10^3 \leq x_1, y_1, x_2, y_2, x_3, y_3 \leq 10^3 \\ \text{$(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are not collinear.} \\ \hline \end{array}\\

&\begin{array}{|c|c|l|} \hline \text{Subtask} & \text{Points} & \text{Constraints} \\ \hline 1 & \mathbf{33} & \text{$y_1 = y_2$ and $x_2 = x_3$} \\ \hline 2 & \mathbf{33} & y_1 = y_2 \\ \hline 3 & \mathbf{34} & \text{No further constraints.} \\ \hline \end{array}\\

\end{align*}$$$$$$

Output

If there are infinitely many completing points, output INFINITE.

Otherwise, if there are finitely many completing points, output each of them in sorted order; output the ones with smaller $$$x$$$-coordinates first, breaking ties by outputting the ones with smaller $$$y$$$-coordinates first.

For each completing point, output five lines: 


point: <x> <y>
area: <a>
is rhombus: <yes/no>
is rectangle: <yes/no>
-------------------------
Replace the values enclosed in angle brackets with the information describing each completing point, as follows:
  • <x> and <y> are the $$$x$$$ and $$$y$$$ coordinates of this completing point. Give each value to exactly two places after the decimal point, truncating the rest.
  • <a> is the area of the formed parallelogram. Again, give the value to exactly two places after the decimal point, truncating the rest.
  • <yes/no> on the third line is yes if the formed parallelogram is a rhombus, and no otherwise
  • <yes/no> on the fourth line is yes if the formed parallelogram is a rectangle, and no otherwise
  • This last line is just a "separator" to easily visually distinguish between different completing points; the - appears 25 times

The checker is strict. Do not even output extra spaces or newlines.

Examples
Input
1 1
2 1
2 3
Output
point: 1.00 -1.00
area: 2.00
is rhombus: no
is rectangle: no
-------------------------
point: 1.00 3.00
area: 2.00
is rhombus: no
is rectangle: yes
-------------------------
point: 3.00 3.00
area: 2.00
is rhombus: no
is rectangle: no
-------------------------
Input
0 0
5 0
-3 4
Output
point: -8.00 4.00
area: 20.00
is rhombus: no
is rectangle: no
-------------------------
point: 2.00 4.00
area: 20.00
is rhombus: yes
is rectangle: no
-------------------------
point: 8.00 -4.00
area: 20.00
is rhombus: no
is rectangle: no
-------------------------