Example 3
Triangle rectangular in (a, b)
(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
a(a-e)+b^2=0
Now we claim circumcenter lies on the horizontal side, say,
T: y=0
> | with(Groebner):H:=Basis({x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2, a*(a-e)+b^2 },tdeg(x,y,e,a,b));
NormalForm(y, H, tdeg(x,y,e,a,b)); |
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(4.1.1) |
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So the theorem is not true. Let us see when this happens, negation of (H ==> T) yields (H & not T).
> | HH:=Basis({x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2, a*(a-e)+b^2, y*t-1 },plex(x,y,t,e,a,b)); |
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(4.1.2) |
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In particular, if theorem is false implies
eb=0
a^2b+b^3=0
a^2-ae+b^2=0
Thus the negation of one of these yields a true statement:
> | HHH:=Basis({x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2, a*(a-e)+b^2, e*b*t-1 },tdeg(x,y,t,e,a,b));
NormalForm(y, HHH, tdeg(x,y,e,a,b)); |
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(4.1.3) |
> | HHHH:=Basis({x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2, a*(a-e)+b^2, (a^2*b+b^3)*t-1 },tdeg(x,y,t,e,a,b));
NormalForm(y, HHHH, tdeg(x,y,e,a,b)); |
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(4.1.4) |
> | H_5:=Basis({x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2, a*(a-e)+b^2, (-a^2+a*e-b^2)*t-1 },tdeg(x,y,t,e,a,b));
NormalForm(y, H_5, tdeg(x,y,e,a,b)); |
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(4.1.5) |
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