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As all these equations are polynomials, you should rather use polynomial methods:

sage: x1, y1, x2, y2=polygens(QQ,'x1,y1,x2,y2')
sage: eq1 = (x1-1)*(x1-2) + (y1-3)*(y1-4)
sage: eq2 = (x2-1)*(x2-2) + (y2-3)*(y2-4)
sage: eq3 = x1+x2 - 3
sage: eq4 = y1+y2 - 7
sage: I = x1.parent().ideal([eq1,eq2,eq3,eq4])
sage: I.variety()

This answers that the zero set has dimension 1, so you cannot hope for a finite set of solutions. Next

sage: I.elimination_ideal([x2,y2])
Ideal (x1^2 + y1^2 - 3*x1 - 7*y1 + 14) of Multivariate Polynomial Ring in x1, y1, x2, y2 over Rational Field

gives you the equation of the circle in the x1,y1 plane that describes the solutions.