Elimination
(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
> | with(PolynomialIdeals): EliminationIdeal(<x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2,y>, {a,b,e});factor(%); |
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(2.2.2.1) |
> | Saturate(<x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2,-e^2*a+e*b^2+e*a^2>,y); |
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(2.2.2.2) |
> | EliminationIdeal(<2*x*e-e^2, 2*x*a-a^2+2*y*b-b^2, e*a^2-e^2*a, b*e>,{a,e}); |
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(2.2.2.3) |
> | factor(%); |
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(2.2.2.4) |
> | EliminationIdeal(<2*x*e-e^2, 2*x*a-a^2+2*y*b-b^2, e*a^2-e^2*a, b*e>,{e,b}); |
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(2.2.2.5) |
> | NormalForm(y,<x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2,-e^2*a+e*b^2+e*a^2, b*e*t-1>, tdeg(x,y,e,t,a,b)); |
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(2.2.2.6) |
> | NormalForm(y,<x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2,-e^2*a+e*b^2+e*a^2, -e*a*(-a+e)*t-1>, tdeg(x,y,e,t,a,b)); |
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(2.2.2.7) |
> | EliminationIdeal(<2*x*e-e^2, 2*x*a-a^2+2*y*b-b^2, e*a^2-e^2*a, b*e>,{a,b}); |
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(2.2.2.8) |