This is an alternate proof of the crossbar theorem, meant to show its close relationship to Pasch's theorem.

Crossbar Theorem (angle version)

Theorem. If the ray A--D-> is between rays A--C-> and A--B->, then A--D-> intersects the segment B-C.

Proof Construct point Q with Q-A-B using Axiom B₂. Then Pasch's Theorem applied to triangle QBC implies thet the line <-A--D-> intersects either segment Q-C or segment B-C. First note that this intersection point must be on the ray A--D->, since it must be on the same side of <-A--B-> as C. It's easy to prove that this intersection point can't be either Q, B, or C. So the only thing to left to rule out is that the intersection point is between Q and C. For this we'll prove a generic Lemma.

Lemma If Q,D o.s. <-A--C->, then A--D-> does not intersect C--Q->.

Proof of Lemma If point F is on A--D-> and C--Q->, then Q,F s.s. <-A--C-> and D,F s.s. <-A--C->, then by Axiom B₄, Q,D s.s. <-A--C->, contradicting the hypothesis. QED

So we need to prove Q,D o.s. <-A--C->. But Q,B o.s. <-A--C-> (since Q-A-B), and D,B s.s. <-A--C-> (since ray A--D-> is between rays A--C-> and A--B->), so Q,D o.s. <-A--C-> by the corollary (c) of Axiom B₄. QED

Uses of the Crossbar Theorem

Very soon we'll use the Crossbar theorem to prove that the diagonals of a convex quadrilateral intersect. But its most important use is in the proof of the Neutral Geometry version of the Exterior Angle Theorem. This theorem is then used repeatedly in triangle studies. Much later the Crossbar Theorem will be used in proving that certain statements are equivalent in Neutral Geometry to Euclid's parallel postulate (e.g. Theorem 8.2.6), still later to prove properties of critical parallels in Hyperbolic Geometry (e.g. Lemma 8.4.7).