*Problem*

*Solution*

To simplify, let i be a generic index from 0,1,2, and j=i+1, k=i-1, both *modulo* 3.

At first sight, we can employ van Obel's theorem:

For, summing for i=0,1,2 we'll get the ratios on the right and their inverses, so that

because for a positive x, x+1/x\ge 2. It follows that at least one of the ratios in question is not less than 2. It does not seem possible to derive from here that at least one of the ratios is not greater than 2. So, we'll have to try another approach.

van Obel's theorem is intimately related to the barycentric coordinates. The three coordinates are defined by the ratios

The sum of the three coordinates is always 1 because, say,

Thus at least one of the coordinates is not greater than 1/3 and at least one of the coordinates is not less than 1/3.

Assume {PQ_1}/{P_1Q_1} ≥ 1/3, then {P_1Q_1}/{PQ_1} = {P_1Q_1}/{PQ_1}=(PP_1 + PQ_1)/PQ_1 ≤ 3, implying {PP_1}/{PQ_1}≤2, as required.