Changing Point of View
-------- ----- -- ----

TeffalHead FatBody has stayed out too late on the planet
BadTrash and is in danger of being consumed by a Larger
BageGarLectorCol.  To get to safety TeffalHead must get
to base A or base B or the ZoomTube that connects them.
He knows his own position, C, and the ZoomTube is a
perfectly straight line between A and B (woe betide a
zoomer in a curved ZoomTube).  TeffalHead needs to know
immediately which he is closest to, A, B, or some point
on the ZoomTube between A and B.

TeffalHead knows the xy-coordinates of points A, B, and
C.  Like any good robotminded soul, he expects to trans-
late and rotate the xy-coordinate system to make a new
x'y'-coordinate system in which A has x'y'-coordinates
(0,0) and B has x'y'-coordinates (L,0), where L is the
distance from A to B.  Then the answer can be easily
read from the x' coordinate of C.

Unfortunately, living up to his first name, which means
`forgetful in emergencies', TeffalHead has forgotten the
program that finds the x'y'-coordinate system.  He as
put out a call for help, and as the only emergency prog-
grammer within range, you must send him a program tout
de suite.

Note you are permitted to translate and rotate the
xy-coordinates, but NOT to reflect across a coordinate
axis.  Unnecessary reflections are a terrible breech
of robot etiquette.  Thus the y' coordinate of C is
unambiguous.

Input
-----

For each of several cases, one line, containing

	Ax Ay Bx By Cx Cy

where the xy-coordinates of points A, B, and C are re-
spectively (Ax,Ay), (Bx,By), and (Cx,Cy).  Input ends
with an end of file.

Output
------

For each case one line containing:

	(Cx',Cy') L ANS

where (Cx',Cy') are the x'y'-coordinates of C, L is
the length of AB, and ANS is one of the following:

	A		If TeffalHead is closest to A.

	B		If TeffalHead is closest to B.

	ZoomTube	If TeffalHead is closest to a
			point on the ZoomTube between
			A and B.

The x'y'-coordinates and L must be accurate to plus or
minus 0.001.

Example Input
------- -----

0 0 1 0 0.5 -6
5.0 3.0 5.5 2.5 5.0 4.0
5.0 3.0 5.5 2.5 5.0 1.0

Example Output
------- ------

(0.500,-6.000) 1.000 ZoomTube
(-0.707,0.707) 0.707 A
(1.414,-1.414) 0.707 B



File:	   changeview.txt
Author:	   Bob Walton <walton@deas.harvard.edu>
Date:	   Sun Oct 26 06:52:33 EST 2003

The authors have placed this file in the public domain;
they make no warranty and accept no liability for this
file.

RCS Info (may not be true date or author):

    $Author: hc3 $
    $Date: 2003/10/26 11:52:04 $
    $RCSfile: changeview.txt,v $
    $Revision: 1.1 $