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

**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 **B _{2}**. 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

**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 **B _{4}**, 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 **B _{4}**. QED

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).

Len M. Smiley Last modified: Mon Feb 5 13:16:10 AKST 2001