= Maximum separation of two circles
{title2=$m(C_1,C_2)=|c_1-c_2|+r_1+r_2$}
For two nonempty planar <circles> with centers $c_i$ and radii $r_i\ge0$, the greatest <Euclidean distance> between their points is the displayed expression. The <triangle inequality> gives the upper bound. When the centers differ, choose the two boundary points outward along the line of centers to attain it; when they coincide, choose opposite radial directions. For any third nonempty <circle> and a point $u$ on it, $|z-w|\le|z-u|+|u-w|$ proves $m(C_1,C_2)\le m(C_1,C_3)+m(C_2,C_3)$. This is not a <metric>, since $m(C,C)=2r$ for a positive-radius <circle>.
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