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.
(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)); |
![]() |
|
![]() |
(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)); |
![]() ![]() |
(4.4.3) |
> |
> |