Solving triangles with trigonometry Time to dig up your old trigonometry notes from high school! The challenge is to solve the unknown sides and angles of different triangles. And as is customary in code golf, the smallest working code wins.

This is not a trivial problem; my reference implementation in python is currently down to 838 837 characters, but I'm sure you'll be able to golf solutions far smaller.

Additionally, if you're stuck, this section on Wikipedia should get you going: Triangle: Computing the sides and angles.

Input

The following triangle shows the names of the sides and angles used in this challenge. Note that sides are lowercase and angles are uppercase.


  


Input is given as six space-separated values, either on stdin or as command-line arguments (your choice). The six values correspond to the sides a, b, c and the angles A, B, C. The unknown sides are given as question marks (?). Both input and output angles has to be in radians. You may assume that the input values are correct (you don't have to validate anything). You may also assume that the input triangle is non-degenerate, and that all sides and angles are nonzero.

The following example input tells you that side a is 8, side b is 12 and angle A is 0.5 radians:

8 12 ? 0.5 ? ?


Output

Output is given in the same format as the input - six space-separated numbers on stdout. The only exception is when it is not possible to solve the input triangle - then the string "No solution" must be written to stdout. If two solutions are possible, they are both outputted with a newline between them.

Following is the output for the above input:

8.0 12.0 16.0899264342 0.5 0.802561439714 1.83903121388
8.0 12.0 4.97205505116 0.5 2.33903121388 0.302561439714


Output is not required to have a lot of precision, but at least a couple decimals is required.

Rules


Input is read from stdin or command-line arguments
Output is written to stdout
If two solutions are possible with the given input, output both
If there is too little information to get one or two clear solutions, consider it a "No solution" case
No built-in or pre-existing code may be used (of course you can use trig functions, but not "solveTriangle" or such)
Shortest code wins


Test cases

In   3 4 5 ? ? ?

Out 3.0 4.0 5.0 0.643501108793 0.927295218002 1.57079630572



In   ? 4 ? 0.64 0.92 1.57

Out 3.00248479301 4.0 5.02764025486 0.64 0.92 1.57



In   ? ? 5 ? 0.92 ?

Out No solution



In   ? ? 5 ? 0.92 1.57

Out 3.03226857833 3.97800936148 5.0 0.65159265359 0.92 1.57



In   8 12 ? 0.5 ? ?

Out (two solutions)

8.0 12.0 16.0899264342 0.5 0.802561439714 1.83903121388
8.0 12.0 4.97205505116 0.5 2.33903121388 0.302561439714




In   8 12 ? ? .5 ?

Out 8.0 12.0 18.3912222133 0.325325285223 0.5 2.31626736837

Good luck!
Python, 441 chars

from math import*
V=[map(float,raw_input().replace('?','0').split())+[0]]
for i in' '*9:
 W=[]
 for a,b,c,A,B,C,R in V:
  if B and C:A=A or pi-B-C
  if a:
   if A:R=R or a/sin(A)
   else:
    if b and c:A=acos((b*b+c*c-a*a)/2/b/c)
    elif R:N=asin(a/R);W+=[(b,c,a,B,C,N,R)];A=pi-N
  else:a=R*sin(A)
  W+=[(b,c,a,B,C,A,R)]
 V=W
V=[T for T in V if all(t&gt;0 for t in T)]
if V:
 for T in V:print' '.join(map(str,T[:-1]))
else:print'No solution'


Does your typical trig to compute the answer.  The current possible solutions are stored as tuples in V.  Any unknown values are recorded as 0.  A seventh variable R is the value a/sin(A)==b/sin(B)==c/sin(C).

I use a trick where the a/b/c values are cycled each iteration to avoid lots of redundant logic.  The inner loop only needs to compute values of the A side or angle.