## Similarity

Consider,in an oriented plane,a direct rectangle AEFD.

Designate by B and C the midpoints of [AE] and [FD] respectively.
Let S be a direct plane similitude that transforms A onto C and E onto B.

1) a- Determine the ratio k and an angle alpha of S.
b- Show that S(F)=E and deduce S(D).
2)Let W be the center of S and let h be the transformation defined by h=SoS.
a- Determine the nature and the characteristic elements of h.
b- Find h(D) and h(F) and construct the point W.
3) Designate by I the midpoint of [BE].
Prove that W,C and I are collinear and express WC(vector) in terms of WI.
4) The complex plane is referred to the orthonormal system (A,u,v) where zB=sqrt(2)
and zD=2i.
Find the complex form of S and determine the affix of W.

My Work :~:
1) a) A ------> C
E ------> B So CB=kAE => ratio(k)= [CB]/[AE] : but AD=CB : k=1/sqrt(2) .

: (AE,CB) = -pi/2 +2kpi .

b) S(F)=E : E -----> B
F -----> E : => BE=kEF => k=BE/EF : but BE=AE/2 = sqrt2 : and EF=AD=2 so k=1/sqrt(2) .

: also , (EF,BE) = -pi/2 +2kpi ~~ So S(F)=E .
Deduce S(D) ~?~?~

2) h(W,-1/2).
I must find S(D) to replace it .
SoS(D) = S(S(D) .....

SoS(F) = S(E) = B .

W is the point of intersection of two circles with diameter D and it's image , diameter [FB].

3) W ,C and I are collinear . . . I think I must find the image of C and it must be I .

4) Z=az + b

a=[1/sqrt2,-pi/2]= -i/sqrt(2)
zB and zD are given but their is no any relation between both .

: : : : : : : : : Thanks in advance : : : : : : : : :