Crossbar Theorem (angle version)

This theorem seems to be stronger than Theorem 3.2.6 in the text. That is, it has a weaker hypothesis (the condition on X ), but the same conclusion. I believe this is the more common statement of the theorem.

Theorem. If X is an interior point of the angle < WUV, then the ray U--X-> intersects the segment W--V at a point Y such that W-Y-V.

Proof. Let Q-U-V (by ruler placement). We apply Pasch's theorem to triangle VQW and line <-U-X->. This tells us only that line <-U-X-> intersects either segment Q--W or segment W--V. But it's easy to show that, first, this intersection point must be on ray U--X-> . (Try it!). Next show that this intersection point can't be on the interior of segment U--W. To do this show that Q,X o.s. UW, and prove that this last fact alone guarantees that ray U--X-> and segment W--Q do not intersect. (Prove this Lemma, or the stronger version with ray W--Q-> in the conclusion). It only remains to rule out the Pasch intersection point being W,V, or Q. (prove each!). QED

Len Smiley