When evaluating poverty, the relative poverty line may be considered as a percentage of the median income or it may be a percentage of the average income. It is proved that, with a poverty line relative to the median income, reducing poverty may become less costly in proportion to the total income as poverty increases (measured by the Sen, the Sen-Shorrocks-Thon or the Foster-Greer-Thorbecke poverty indexes) by passing from a Lorenz concentration curve to another curve associated with more poverty. This is obviously a paradox, although a largely overlooked one. However, it is shown that the paradox vanishes if the poverty line is relative to the average income. The demonstration is both experimental (algebraic or numeric where necessary) based on families of non-intersecting concentration curves produced by various algebraic functions (power function, exponential function or elliptic function but also produced by the Pareto distribution) and it is analytic. One concludes that the poverty line should be relative to the average income rather than to the median income.