Plane Geometry - Bruce E. Shapiro

Section 17The Crossbar TheoremTheorem 17.1 (The Crossbar Theorem) Let △ABC be a triangle andD a point in the interior of angle ∠BAC. Then there is a point E that lieson both −→ AD and BC.Figure 17.1: Illustration of the crossbar theorem (theorem 17.1). If D is inthe interior of ∠CAB then it must cross side BC of △ABCProof. The proof will apply the z-theorem three times.Choose points P and Q such that P ∗ A ∗ D and Q ∗ A ∗ C as shown infigure 17.2. Define l = ←→ P D.Since D is in the interior of ∠BAC neither B nor C lie on l.We can apply Pasch’s Theorem (theorem 15.12) to △BCQ. Since ADcrosses CQ of △BCQ it must cross either BQ or BC. This gives us four87