Consider,in an oriented plane,a direct rectangle AEFD.
such that (AE,AD)=pi/2+2kpi , AE=2sqrt(2) and AD=2.
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)
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) ~?~?~
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
zB and zD are given but their is no any relation between both .
: : : : : : : : : Thanks in advance : : : : : : : : :