with(Groebner);

Proof that the medians of a triangle all meet at a single point.

The vertices are indep[endent: A = (0,0), B = (u1,0), C = (u2,u3)

and the midpoints are: c = (x1,0) = (u1/2,0), a = (x2,x3) = ((u1+u2)/2,u3/2)

and b = (x4,x5) = (u2/2,u3/2)

with(Ore_algebra);

S := poly_algebra(y,u1,u2,u3,x1,x2,x3,x4,x5,x6,x7,x8,x9);

T := termorder(S,plex(y,u1,u2,u3,x1,x2,x3,x4,x5,x6,x7,x8,x9));

c1 := 2*x1 - u1;

a1 := 2*x2 - u1 - u2;

a2 := 2*x3 - u3;

b1 := 2*x4 - u2;

b2 := 2*x5 - u3;

Next, we set up the intersection point (x6,x7) = Aa.Bb

I1 := x9*x2 - x8*x3;

I2 := x9*(x4-u1) - x5*(x8-u1);

And the intersection point (x8,x9) = Aa.Cc

J1 := x7*x2 - x6*x3;

J2 := x7*(u2-x1) - u3*(x6-x1);

The Conclusion is, quite simply, that (x6,x7) = (x8,x9)

C1 := x9-x7;

C2 := x8-x6;

Check := [a1,a2,b1,b2,c1,c2,I1,I2,J1,J2,1-y*C1];

gbasis(Check,T);

Check := [a1,a2,b1,b2,c1,c2,I1,I2,J1,J2,1-y*C2];

gbasis(Check,T);

YEAH!