lib to do advanced geometry calculation?

[oyster]

My problem is some complex, because some condition is not supplied as a
given value
please have a look at
http://www.lusiya.com/bbs/attachment/thumb/Mon_0906/80_3201_d2fa7bd75d28675.jpg
because I have to do futher calculation based on previous steps as
http://www.lusiya.com/bbs/attachment/thumb/Mon_0906/80_3201_d342e52235049e0.jpgshows,
I want to get a precise value if possible, that is to say, (sqrt(2),
0), but not (1.414.0)

[description]
(clear condition)
A=point(0,1)
B=point(1,1)
C=point(0,0)
D=point(1,0)
AC=segment(A, C)
BD=segment(B, D)
CD=segment(C, D)
AB=segment(A, B)

I=midpoint(B, D)

(vague condition)
E=pointOn(AC)
F=pointOn(BD)
EF=segment(E, F)

G=mirror(A, EF)
G=pointOn(CD)

H=mirror(I, EF)
H=pointOn(AB)


(solve)
E
F

[/description]

Above is the description of my question, but I also hope I can write a
problem like that to solve it.
The only lib I know is sympy, but it seems that I must write different and
long code accroding to different problem

So is there any advanced (and lazy of cause) lib to handle my question?

thank you
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