restart;  
with(geometry):  
with(plots):  
_EnvHorizontalName = 'x':  _EnvVerticalName = 'y':
point(A, -1, 9):                                                                                                       
point(B, -5, 0):
point(C, 6, 0):
triangle(ABC,[A,B,C]):
midpoint(M1,A,C): midpoint(M2,B,C):midpoint(M3,A,B):
rotation(J, C, Pi/2, 'counterclockwise', M1):triangle(AJC,[A,J,C]):
rotation(Ii, C, Pi/2, 'counterclockwise', M2):triangle(BIC,[B,Ii,C]):
rotation(K, A, Pi/2, 'counterclockwise', M3):triangle(AKB,[A,K,B]):
midpoint(O1,K,J): coordinates(O1):
midpoint(O2,A,Ii): coordinates(O2):  
poly:=[coordinates(A),coordinates(J),coordinates(Ii),coordinates(K)]:   

display(draw([A(color = black, symbol = solidcircle, symbolsize = 12), 
B(color = black, symbol = solidcircle, symbolsize = 12), 
C(color = black, symbol = solidcircle, symbolsize = 12), 
J(color = black, symbol = solidcircle, symbolsize = 12), 
polygonplot(poly,color = "DarkGreen", transparency = 0.5),
ABC(color = red ),
BIC(color = green),
AKB(color = grey),
AJC(color =blue)]),
textplot([[coordinates(A)[], "A"],[coordinates(J)[], "J"],[coordinates(Ii)[], "I"],   
[coordinates(B)[], "B"], [coordinates(K)[], "K"], 
[coordinates(C)[], "C"]], 
align = [above, right]),  axes = none);
Error, (in geometry:-draw) the option must be of type equation or name. I don't see how to correct this error/
 

what is the homology matrix that plates the ABCE square on the NPCM square
I think it may bi find out with the rotation angle, the vector of translation and the homothety ratio.
restart;  
with(geometry):  
with(plots):  
_EnvHorizontalName = 'x':  _EnvVerticalName = 'y':

point(A, 0, 1):
point(B, 1, 1):
point(C, 1, 0):
point(E, 0, 0):
square(Sq, [A, B, C, E]):
Phi := (1 + sqrt(5))/2:
point(N, (2 - Phi)/(Phi - 1), 1):
line(BE, [B, E]):
MakeSquare(s1, [N, C, 'diagonal']):
point(M, (3 - sqrt(5))/(2*sqrt(5) - 2), (3 - sqrt(5))/(2*sqrt(5) - 2)):
point(P, (1 + sqrt(5))/(2*sqrt(5) - 2), (3*sqrt(5) - 5)/(2*sqrt(5) - 2)):
T:=<simplify(coordinates(midpoint(O1,E,B))-coordinates(midpoint(O2,M,P)))>:
simplify(distance(O1,O2)):
line(MN,[M,N]):eq:=Equation(%,[x,y]):sol:=solve(eq,y):
Ang:=Pi/2-arctan(diff(sol,x)):
r:=simplify(distance(N,M)):
line(MP,[M,P]):eq:=Equation(%,[x,y]):subs(y=0,%):point(Q,solve(%,x),0):
line(PQ,[P,Q]):
homology(Sq1, Sq, C, Ang, 'clockwise', r):

display(draw([A(color = black, symbol = solidcircle, symbolsize = 12), 
B(color = black, symbol = solidcircle, symbolsize = 12), 
C(color = black, symbol = solidcircle, symbolsize = 12), 
E(color = black, symbol = solidcircle, symbolsize = 12), 
N(color = black, symbol = solidcircle, symbolsize = 12), 
Sq(color = red, filled = true, transparency = 0.9), 
BE(color = green), 
PQ(color = black),
 Sq1(color = black), 
s1(color = red, filled = true, transparency = 0.8)]), 
textplot([[coordinates(A)[], "A"], 
[coordinates(B)[], "B"], 
[coordinates(E)[], "E"], 
[coordinates(N)[], "N"], 
[coordinates(P)[], "P"], 
[coordinates(M)[], "M"], 
[coordinates(Q)[], "Q"], 
[coordinates(C)[], "C"]], 
align = [above, right]), view = [-0.6 .. 1.5, 0 .. 1], axes = none);
 

How to find the similarity matrix that applies A in N, B in P, C in C and B in M;
 

restart;  
with(geometry):  
with(plots):  
_EnvHorizontalName = 'x':  _EnvVerticalName = 'y':
#Vdot := proc(U, V) local i; add(U[i]*V[i], i = 1 .. 2); end proc
;

with(LinearAlgebra):
point(A, 0, 1);
point(B, 1, 1);
point(C, 1, 0);
point(E, 0, 0);
square(Sq, [A, B, C, E]);
Phi := (1 + sqrt(5))/2;
point(N, (2 - Phi)/(Phi - 1), 1);
line(BE, [B, E]);
MakeSquare(s1, [N, C, 'diagonal']);
point(M, (3 - sqrt(5))/(2*sqrt(5) - 2), (3 - sqrt(5))/(2*sqrt(5) - 2));
point(P, (1 + sqrt(5))/(2*sqrt(5) - 2), (3*sqrt(5) - 5)/(2*sqrt(5) - 2));
                               A

                               B

                               C

                               E

                               Sq

                             1   1  (1/2)
                      Phi := - + - 5     
                             2   2       

                               N

                               BE

                               s1

                               M

                               P

 display(draw([
A(color = black, symbol = solidcircle, symbolsize = 12),   
B(color = black, symbol = solidcircle, symbolsize = 12),   
C(color = black, symbol = solidcircle, symbolsize = 12),    
E(color = black, symbol = solidcircle, symbolsize = 12), 
N(color = black, symbol = solidcircle, symbolsize = 12 ),  
Sq(color=red),BE(color=green),  
s1(color = blue)]),   
textplot([[coordinates(A)[], "A"],   
[coordinates(B)[], "B"], 
[coordinates(E)[], "E"], 
[coordinates(N)[], "N"],
[coordinates(P)[], "P"],
[coordinates(M)[], "M"],   
[coordinates(C)[], "C"]], align = [above, right]), axes = none); Thank you.

S1 := 441;
S2 := 1109;
S3 := 511;
S4 := 900;
S5 := 2904;
S6 := 285;

S2 + S3 + S4 + S5 + S6;                             
for i to 5 do
    S*[i] + S[i + 1];
end do;
Thank you.

restart;
with(geometry):
with(plots):
_EnvHorizontalName = 'x':
_EnvVerticalName = 'y':
point(A, [-2, 7]):
point(B, [-5, -2]):
point(C, [8, -2]):
triangle(ABC, [A, B, C]):
display([draw[A(color = black, symbol = solidcircle, symbolsize = 6), 
               B(color = black, symbol = solidcircle, symbolsize = 6), 
               C(color = black, symbol = solidcircle, symbolsize = 6), 
               ABC(color = blue)], 
         textplot([[coordinates(A)[], "A"], 
              [coordinates(B)[], "B"], 
              [coordinates(C)[], "C"]],
              align = [above, right])], 
              axes = none);
Would you explain to me why this error ? Thank you.
Error, (in plots:-display) expecting plot structures but received: [geometry:-draw[A(color = black, symbol = solidcircle, symbolsize = 6), B(color = black, symbol = solidcircle, symbolsize = 6), C(color = black, symbol = solidcircle, symbolsize = 6), ABC(color = blue)]]