  In studying properties of simple drawings of the complete graph in the
sphere, two natural questions arose for us: can an edge have multiple segments
on the boundary of the same face? and is each face the intersection of sides of
3-cycles? The second is asserted to be obvious in two previously published
articles, but when asked, authors of both papers were unable to provide a
proof. We present a proof. The first is quite easily proved and the technique
yields a third, even simpler, fact: no three edges at a vertex all have
internal points incident with the same face.
