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. Lemma, are 3 distinct nonzero points in the plane then the 6 points p, q, r, 2p, 2q, 2r are not convexly independent. This is clear from a picture and can be proved for instance by considering the 4 points 0, p, q, r. There are two cases. 1. If these 4 points are convexly independent, assume for instance that pq is a diagonal of the quadrangle 0prq, p.2