Plane Geometry - Bruce E. Shapiro

90 SECTION 17. THE CROSSBAR THEOREMTheorem 17.2 (MacLane’s Continuity Axiom a ) A point D is in theinterior of angle ∠BAC if and only if the ray −→ AD intersects the interior ofthe segment BC.a MacLane [MacLane, 1959] takes this result as an axiom and uses this name because itimplies Birkhoff’s [Birkhoff, 1932] continuity axiom (theorem 20.2).Figure 17.6: Illustration of MacLane’s Continuity Axiom (theorem 17.2.)Proof. (⇒) Suppose that D is in the interior of ∠BAC.−→crossbar theorem, AD intersects BC.Then by theSince the intersection point lies on a ray that is interior to ∠BAC, it doesnot lie on an endpoint of BC (definition of interior; else it would lie on one ofthe two rays that define the angle, not the intersection of their half-planes);hence the intersection point is interior to BC.(⇐) Suppose that −→ AD intersects the interior of BC. Call the point ofintersection E.−→ −→ −→ −→ −→Hence B ∗ E ∗ C. By theorem 16.8, AB ∗ AE ∗ AC. Hence AE = AD is inthe interior of angle ∠BAC. Hence D is in the interior of ∠BAC.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012