  Circuits and extended formulations are classical concepts in linear
programming theory. The circuits of a polyhedron are the elementary difference
vectors between feasible points and include all edge directions. We study the
connection between the circuits of a polyhedron $P$ and those of an extended
formulation of $P$, i.e., a description of a polyhedron $Q$ that linearly
projects onto $P$.
  It is well known that the edge directions of $P$ are images of edge
directions of $Q$. We show that this `inheritance' under taking projections
does not extend to the set of circuits. We provide counterexamples with a
provably minimal number of facets, vertices, and extreme rays, including
relevant polytopes from clustering, and show that the difference in the number
of circuits that are inherited and those that are not can be exponentially
large in the dimension.
  We further prove that counterexamples exist for any fixed linear projection
map, unless the map is injective. Finally, we characterize those polyhedra $P$
whose circuits are inherited from all polyhedra $Q$ that linearly project onto
$P$. Conversely, we prove that every polyhedron $Q$ satisfying mild assumptions
can be projected in such a way that the image polyhedron $P$ has a circuit with
no preimage among the circuits of $Q$. Our proofs build on standard
constructions such as homogenization and disjunctive programming.
