Lemma

Let $A = (a, a')$, $B = (b, b')$, $C = (c, c')$ be three points in $\mathbf{R}^2$ such that $a < b < c$. The following statements are equivalent:

a) B is below AC; b) C lies above the line passing through A and B; c) A is above the line passing through B and C; d) $p(AB) \leq p(AC)$; e) $p(AC) \leq p(BC)$.

The lemma still holds when one replaces "above" (resp. "below") by "strictly above" (resp. "strictly below") and the sign $\leq$ by < (fig. 1).

!Graph showing points A, B, C, and segments labeled with x values a, b, c

Fig. 1