  A relation extends another relation consistently if its symmetric,
respectively its asymmetric, part contains the corresponding part of the
smaller relation. It is shown that there exists no finite circular chain made
from two transitive relations $\mathbf{A}$ and $\mathbf{B}$ with at least one
link from their asymmetric parts if and only if there exists a total preorder
which consistently extends both. Additionally, this extension is uniquely
determined if and only if the reflexive transitive closure of the union of
$\mathbf{A}$ and $\mathbf{B}$ is total. Applications: (1) If the steps of a
walk come from two positive cones, with at least one step from one of the
cones' non-linear parts, then returning to the origin is impossible if and only
if there exists a third cone of which the linear part contains each of the
linear parts of the two original cones, and of which the non-linear part
contains each of the two non-linear parts. (2) Reminiscent of the Fundamental
Theorem of Asset Pricing, absence of arbitrage is equivalent to the existence
of a complete preference order which consistently extends a market's fair
exchange relation and its objective strict preference order. (3) Another
impossibility in microeconomics: For two agents, or an agent and a market, with
additive positively homogeneous preferences, an allocation can never be Pareto
optimal if they are not `cut from the same cloth'.
