Lost in Space
Time Limit: 1 Second Memory Limit: 32768 KB
William Robinson was completely puzzled in the music room; he could not find
his triangle in his bag. He was sure that he had prepared it the night before.
He remembered its clank when he had stepped on the school bus early that morning.
No, not in his dream. His triangle was quite unique: no two sides had the same
length, which made his favorite peculiar jingle. He insisted to the music teacher,
Mr. Smith, that his triangle had probably been stolen by those
aliens and thrown away into deep space.
Your mission is to help Will find his triangle in space. His triangle has been
made invisible by the aliens, but candidate positions of its vertices are somehow
known. You have to tell which three of them make his triangle. Having gone through
worm-holes, the triangle may have changed its size. However, even in that case,
all the sides are known to be enlarged or shrunk equally, that is, the transformed
triangle is similar to the original.
Input
The very first line of the input has an integer which is the number of data
sets. Each data set gives data for one incident such as that of William. At
least one and at most ten data sets are given.
The first line of each data set contains three decimals that give lengths of
the sides of the original triangle, measured in centimeters. Three vertices
of the original triangle are named P, Q, and R. Three decimals given in the
first line correspond to the lengths of sides QR, RP, and PQ, in this order.
They are separated by one or more space characters.
The second line of a data set has an integer which is the number of points in
space to be considered as candidates for vertices. At least three and at most
thirty points are considered.
The rest of the data set are lines containing coordinates of candidate points,
in light years. Each line has three decimals, corresponding to x, y, and z coordinates,
separated by one or more space characters. Points are numbered in the order
of their appearances, starting from one.
Among all the triangles formed by three of the given points, only one of them
is similar to the original, that is, ratios of the lengths of any two sides
are equal to the corresponding ratios of the original allowing an error of less
than 0.01 percent. Other triangles have some of the ratios different from the
original by at least 0.1 percent.
The origin of the coordinate system is not the center of the earth but the center
of our galaxy. Note that negative coordinate values may appear here. As they
are all within or close to our galaxy, coordinate values are less than one hundred
thousand light years. You don't have to take relativistic effects into account,
i.e., you may assume that we are in a Euclidean space. You
may also assume in your calculation that one light year is equal to 9.461x10^12
kilometers.
A succeeding data set, if any, starts from the line immediately following the
last line of the preceding data set.
Output
For each data set, one line should be output. That line should contain the point numbers of the three vertices of the similar triangle, separated by a space character. They should be reported in the order P, Q, and then R.
Sample Input
2 50.36493 81.61338 79.96592 5 -10293.83 -4800.033 -5296.238 14936.30 6964.826 7684.818 -4516.069 25748.41 -27016.06 18301.59 -11946.25 5380.309 27115.20 43415.93 -71607.81 11.51547 13.35555 14.57307 5 -56292.27 2583.892 67754.62 -567.5082 -756.2763 -118.7268 -1235.987 -213.3318 -216.4862 -317.6108 -54.81976 -55.63033 22505.44 -40752.88 27482.94
Sample Output
1 2 4 3 4 2Submit
Source: Asia 2000, Tsukuba (Japan)