Counterexample 

 

 

The collection of points 

 

> restart:
 

 

 

> A:=[0,0];B:=[1,0];C:=[0,alpha];DD:=[infinity , 0] ; EE:=[-3*alpha+1,-1];F:=[-2*alpha,alpha];  G:=[2,-1]; H:=[0,1];
 

 

 

 

 

 

 

 

[0, 0]
[1, 0]
[0, alpha]
[infinity, 0]
[`+`(`-`(`*`(3, `*`(alpha))), 1), -1]
[`+`(`-`(`*`(2, `*`(alpha)))), alpha]
[2, -1]
[0, 1] (4.3.1.1)
 

> alpha:=I/sqrt(3);
 

`*`(`*`(`/`(1, 3), `*`(I)), `*`(`^`(3, `/`(1, 2)))) (4.3.1.2)
 

> with(linalg):matrix(3,3,[1,F[1],F[2],1,H[1],H[2],1,EE[1],EE[2]]);
 

Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mn( (4.3.1.3)
 

> L1:=det(%);
 

0 (4.3.1.4)
 

> matrix(3,3,[1,B[1],B[2],1,C[1],C[2],1,EE[1],EE[2]]);
 

Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mn( (4.3.1.5)
 

> L2:=det(%);
 

0 (4.3.1.6)
 

> matrix(3,3,[1,F[1],F[2],1,G[1],G[2],1,A[1],A[2]]);
 

Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mn( (4.3.1.7)
 

> L3:=det(%);
 

0 (4.3.1.8)
 

>
 

>
 

 

 

verify the hypothesis  but not the conclusion of Maclane's theorem.