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Lemma 1b

The perpendicular to ab at b is a supporting line of B, and a and B are on different sides relative to that line.

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Proof :

Lemma 1b is the obvious counterpart of lemma 1a, and will be proved using the same arguments. If b is the closest point of B from a, then a circle C of radius ab centered at a contains only one point of B, namely b.  The tangent to C is thus a supporting line of B.

Similarly to the second part of lemma 1a, if some points of B are on the same side of P than A, then the support point is not the closest point of B from a.  This is opposed to the definition Hausdorff distance.