Let $ABC$ be a triangle. Pick $P$, $Q$, $R$ on sides $BC$, $CA$, $AB$, respectively, and then points $S$, $T$, $U$ on the sides $QR$, $RP$, $PQ$ of triangle $PQR$, respectively. Consider the three triples of lines $AP$, $BQ$, $CR$; $PS$, $QT$, $RU$; $AS$, $BT$, $CU$. If any two of the triples are concurrent, then so too is the third.
If points are allowed to be picked on the sides extended, then the statement needs to be modified to refer to concurrent or parallel lines (unless you're working in the projective plane).
I happened upon this fact accidentally in trying to prove something else. It's not terribly difficult to prove using Ceva's theorem and its converse, but the proof did require some amount of work (for me). The statement of the theorem seems so attractive that I imagine it must have received attention at some point.
Is this theorem well known? And is there some theory within which it could be considered obvious or can be given a purely conceptual proof without calculation? (Or is it plain obvious and I've just missed it?)
This is the Cevian nest theorem, discussed in this paper.
It now has synthetic proofs, but was already known at least in 1886, in Neue merkwürdige Punkte des Dreiecks, Johann Döttl, p. 29.