= Constructible circle
A <constructible circle> has a <constructible point> as its <centre> and <radius> equal to the <distance> between two <constructible points>. When those coordinates lie in a real <field> $F$, its equation $(x-a)^2+(y-b)^2=r^2$ has <coefficients> in $F$ because the squared <radius> is a sum of squared coordinate differences. Intersecting it with a <constructible line> reduces to a <quadratic equation>. Intersecting two distinct <circles> reduces to the same situation by subtracting their normalized equations to obtain the radical axis. Thus each isolated real <intersection> requires at most a real quadratic <field extension>.
Back to article page
