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];

> alpha:=I/sqrt(3);

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

Warning, the protected names norm and trace have been redefined and unprotected

> L1:=det(%);

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

> L2:=det(%);

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

> L3:=det(%);

>

>

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