Yes, we can construct an equilateral triangle ABC with the given conditions using rotations. Starting from any one of the three parallel lines and rotating the other two lines by 60 degrees each will yield the other two points of the triangle, allowing its construction.

1. ROTATION

6. 1. ROTATIONJ FORMULA OVER O(0,0) .
Let A(x,y) any point in plane V ang A’(x’,y’)is
image of A over R 0, , or A’ = R 0, (A).
Let m(XOA)=  .
We have x =OA cos dan y = OA sin and
x’ = OA’ cos (+)
= OA (cos cos - sin sin)
= x cos  - y sin 

7. A’(x’,y’)
A(x,y)


(0,0)

8. .
• y’ = OA’ sin (+)
• = OA(sin  cos  + cos  sin )
• = x sin  + y cos 
• so
• x’ = xcos  - y sin 
• y’ = x sin  + y cos 
• or
 x' cos   sin    x 

 y ' sin   y
   cos    

9. C’(x’,y’)=C’(x*’,y*’)
y*
C(x,y)=C(x*,y*)
(a,b)
x*
(0,0) X
Y

10. 2. ROTATION OVER P(a,b)
• Let we have coordinate system with
centre P(a,b)and has two axis X*
and Y*, X//X* and Y//Y*.
• If C(x*,y*) and C’=RP,(C), then C’
(x*’,y*’) , we have a relation :
 x*'  cos   sin    x *

 y*' sin 
   cos    y *
 

11. In coordinate of X , Y axis , we have :
 x'a  cos   sin    x  a 

 y 'b sin  cos    y  b
    
 x' cos   sin    x   p 

 y ' sin  cos    y  q  
      
p  a cos   b sin   a
q  a sin   b cos   b

12. THEOREM
• Rotation RP, can represent in
composition of two lines
reflection over s and t with P is
(s,t) and m(<(s,t))=½ .
• Rotation is an isometry
• composition of two lines
reflection :
 SAB ,if s//t
M t Ms  
 R P,θ , if t and s not paralel

14. A”
t
T
A’
Q s
P A

15. Theorem
1  R
R P, P, -
Theorem
R P, R P,   R P,  

17. •If s perpendicular to t and P=(s,t) ,
•then MtMs=HP.
s
A”
P E
t
A D A’

18. .
For every line a,b with
• Teorema
a//b, then MbMa=SCD with |CD|=2 x
distance (a,b) and CD a.
P’
P P’’
B
A
D
a
P b

19. Mb Ma = Mb I Ma
= Mb (MsMs )Ma
= Mb MsMs Ma
= (Mb Ms )(Ms Ma)
= HBHA
= SCD with |CD|= 2 |AB|

20. •Translation SAB can represent as
composition of two reflection Ms dan Mt
with s//t and s  AB, and distance of
(s,t) is ½ |AB|.
t
B
s
A

21. • Given three paralel lines a, b dan c.
• Construct an equilateral triangle ABC
with condition A on a, B on b and C on c.
a
b
c

22. Contoh permasalahan
a
A
b
B
c
C

23. • a. Fixed any point A on a.
• b. Rotated line c, with angle 60o over A, we got c’.
• c. Intersection of line c’ and line b, ( c’,b) is point
B.
• d. We can construct equilateral triangle ABC.
•
• We can also start with fixed point B on b or C on
c.
• Can do it ?

24. • a. Fixed any point B on b.
• b. Rotated line a, with angle 60o over B, we got a’.
• c. Intersect of line a’ and line c, ( a’,c) is point C .
• d. We can construct equilqteral triangle ABC.
•
• We can also start with fixed point A on a or C on
c.
• Can do it ?