Plane Geometry - Bruce E. Shapiro

SECTION 17. THE CROSSBAR THEOREM 89Hence BC ∩ −→ AP = ∅. This eliminates equation 17.3 (see figure 17.4).Figure 17.4: Elimination of equation 17.3 by the Z-theorem (heavy lines).Since Q ∗ A ∗ C, Q and C are on opposite sides of ←→ AB.Since C and D are on the same side of ←→ AB (because D is in the interiorof ∠CAB), then Q and D are on opposite sides of ←→ AB (plane separationpostulate).By the Z-theorem −→ −→−→BQ ∩ AD = ∅. Hence BQ ∩ AD = ∅. This elimninatesequation 17.2 (see figure 17.5).Figure 17.5: Elimination of equation 17.2 by the Z-theorem (heavy lines).Since equations 17.2, 17.3, and 17.4 have been eliminated, then equation17.1 must be true.Since ←→ AD intersects BC they have a unique point of intersection. Call thispoint E. This is the point that the theorem says exists.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.