3577 -- Global Positioning System

Language:

Global Positioning System

Time Limit: 1000MS		Memory Limit: 65536K
Total Submissions: 1295		Accepted: 106

Description

Xed has a great interest in Global Positioning System. To simulate it, he constructs a Cartesian coordinate system in a infinite plane whose one unit equals 1 kilometer. Then Xed puts three sensors on three different points (x₁,y₁), (x₂,y₂), (x₃,y₃) and a signal generator in the plane.

All three sensors are linked to a timer. Once a sensor has received the signal it will immediately trigger the timer which records the time. The timer starts to count when it first receives a trigger and records the earliest time when the sensors receive a signal.

Now you are given:

positions of the sensors: (x₁,y₁), (x₂,y₂), (x₃,y₃)
time records: t₁, t₂, t₃seconds. (t₁, t₂, t₃ ≥ 0, and at least one equals 0 which presents the first sensor triggers the timer)
Velocity of the signal: C meters per second (C > 0)

Please help Xed to determine the position of the signal generator. It is guaranteed that such point always exists and is unique.

Input

The input consists of several test cases. Each test case has the following format:
C
x₁ y₁ x₂ y₂ x₃ y₃
t₁ t₂ t₃
The input is terminated by C = 0

Output

For each test case, print the case number on the first line and the position of the generator (x, y) on the second line. x and y should be separated by a single space and should be rounded to six digits to the right of the decimal point. See the sample output below for more details.
　

Sample Input

1000
0 0 1 0 2 1
0 0.414213562373 1

1000
0 1 1 1 2 1
0 0.6 1.6

1000
0 0 0 1 1 0
0.4142135 0 0
1000
0 0 0 -1 0 1
0 0 1

1000
0 0 0 1 0 -1
0 1 0
1000
0 0 1 0 -1 0
0 1 0
1000
0 0 -1 0 1 0
0 0 1
100
0 0 0 1 1 0
0 10 10
0

Sample Output

Case 1:
0.000000 1.000000
Case 2:
0.200000 1.000000
Case 3:
1.000000 1.000000
Case 4:
0.000000 -0.500000
Case 5:
0.000000 -0.500000
Case 6:
-0.500000 0.000000
Case 7:
-0.500000 0.000000
Case 8:
0.000000 0.000000

Source

POJ Founder Monthly Contest – 2008.04.13, Lei Tao