Plane Geometry - Bruce E. Shapiro

SECTION 34. THE PARALLEL PROJECTION THEOREM 175Figure 34.2: Parallel Projection Theorem (theorem 34.2).By the lemma,A ′ i A′ i+1A ′ C ′ = A iA i+1AC= 1 qThus A ′ B ′ = p × A ′ iA ′ i+1 = p × A′ C ′hence A′ B ′q A ′ C ′ = p q = AB , proving theACtheorem in the case where AB/AC is rational.To prove the theorem when x = AB/AC is irrational, define y = A ′ B ′ /A ′ C ′ .Let r be any rational number such that 0 < r < x and define D ∈ t suchthat AD/AC = r (see figure 34.4). Construct m ′ ‖ l through D and defineD ′ as the intersection of m ′ and t ′ . By the first caseSince l ‖ m ‖ m ′ , A ′ ∗ D ′ ∗ B ′ , henceA ′ D ′A ′ C ′ = rr < A′ B ′A ′ C ′ = yHence for every r < x, we have r < y. By a similar argument, for everyr < y, we have r < x. Hence x = y by the comparison theorem for realnumbers (theorem 6.5), x = y.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.