Elimination 

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 

 

 

 

> 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(%);
 

 

POLYNOMIALIDEAL(`+`(`-`(`*`(`^`(e, 2), `*`(a))), `*`(e, `*`(`^`(b, 2))), `*`(e, `*`(`^`(a, 2)))), characteristic = 0, variables = {e, a, b}, known_groebner_bases = (table( [( tdeg(b, a, e) ) = [[1, `*...
POLYNOMIALIDEAL(`+`(`-`(`*`(e, `*`(`+`(`*`(e, `*`(a)), `-`(`*`(`^`(b, 2))), `-`(`*`(`^`(a, 2)))))))), characteristic = 0, variables = {e, a, b}, known_groebner_bases = (table( [( tdeg(b, a, e) ) = [[1... (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);
 

POLYNOMIALIDEAL(`+`(`*`(2, `*`(x, `*`(e))), `-`(`*`(`^`(e, 2)))), `+`(`*`(2, `*`(x, `*`(a))), `-`(`*`(`^`(a, 2))), `*`(2, `*`(y, `*`(b))), `-`(`*`(`^`(b, 2)))), `+`(`*`(e, `*`(`^`(a, 2))), `-`(`*`(`^`... (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});
 

POLYNOMIALIDEAL(`+`(`*`(e, `*`(`^`(a, 2))), `-`(`*`(`^`(e, 2), `*`(a)))), characteristic = 0, variables = {e, a}, known_groebner_bases = (table( [( tdeg(a, e) ) = [[1, `*`(e, `*`(`^`(a, 2))), `+`(`*`(... (2.2.2.3)
 

> factor(%);
 

POLYNOMIALIDEAL(`+`(`-`(`*`(e, `*`(a, `*`(`+`(`-`(a), e)))))), characteristic = 0, variables = {e, a}, known_groebner_bases = (table( [( tdeg(a, e) ) = [[1, `*`(e, `*`(`^`(a, 2))), `+`(`*`(e, `*`(`^`(... (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});
 

POLYNOMIALIDEAL(`*`(b, `*`(e)), characteristic = 0, variables = {e, b}, known_groebner_bases = (table( [( tdeg(b, e) ) = [[1, `*`(b, `*`(e)), `*`(b, `*`(e))]] ] ))) (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));
 

0 (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));
 

0 (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});
 

POLYNOMIALIDEAL(characteristic = 0, variables = {a, b}, known_groebner_bases = (table( [( tdeg(y, x, a) ) = [], ( tdeg(y, x) ) = [], ( tdeg(b, a) ) = [], ( tdeg(a, y, x) ) = [] ] ))) (2.2.2.8)