absolutely... 

 

The student is doing home work. Vaguely remembers? that under some conditions on a  triangle, the circumcenter lies on one side. Asks the computer, to check it....by stating it is always true. 

Image 

 

(x,y) circumcenter, ie.

H:
       x^2+y^2-(x-e)^2-y^2=0
       x^2+y^2-(x-a)^2-(y-b)^2=0
 

       

Now we claim circumcenter lies on the horizontal side, say,

T
: y=0 

 

 

> restart:with(Groebner):H:=Basis({x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2 },tdeg(x,y,e,a,b));
NormalForm(y, H, tdeg(x,y,e,a,b));
 

 

[`+`(`*`(2, `*`(x, `*`(a))), `-`(`*`(`^`(a, 2))), `*`(2, `*`(y, `*`(b))), `-`(`*`(`^`(b, 2)))), `+`(`*`(2, `*`(x, `*`(e))), `-`(`*`(`^`(e, 2)))), `+`(`*`(`^`(e, 2), `*`(a)), `-`(`*`(e, `*`(`^`(a, 2)))...
y (4.4.1)
 

(4.4.2)
 

>
 


So, the theorem is not true...Moreover, for most values of a, b, e, there are values of (x, y) satisfying the hypotheses and not the thesis. 

 

> Basis({x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2, y*t-1}, plex(x,y,t,e,a,b));
 

[`+`(`*`(2, `*`(e, `*`(b))), `*`(`^`(e, 2), `*`(a, `*`(t))), `-`(`*`(t, `*`(e, `*`(`^`(a, 2))))), `-`(`*`(t, `*`(e, `*`(`^`(b, 2)))))), `+`(`*`(`^`(e, 2), `*`(a)), `-`(`*`(e, `*`(`^`(a, 2)))), `*`(2, ...
[`+`(`*`(2, `*`(e, `*`(b))), `*`(`^`(e, 2), `*`(a, `*`(t))), `-`(`*`(t, `*`(e, `*`(`^`(a, 2))))), `-`(`*`(t, `*`(e, `*`(`^`(b, 2)))))), `+`(`*`(`^`(e, 2), `*`(a)), `-`(`*`(e, `*`(`^`(a, 2)))), `*`(2, ...
(4.4.3)
 

>
 

>