Plane Geometry - Bruce E. Shapiro

SECTION 38. THE PYTHAGOREAN THEOREM 193m(∠C) = 90,m(∠A) + m(∠B) = 90Also by the angle sum theorem,m(∠A) + m(∠ACD) = 90m(∠B) + m(∠BCD) = 90Hencem(∠A) = m(∠BCD)m(∠B) = (∠ACD)Thus△ABC ∼ △ACD ∼ △CBDBy the fundamental theorem of similar triangles,HenceAdding,pb = b c and q a = a cb 2 = pc and a 2 = qca 2 + b 2 = pc + qc = c(p + q) = c 2Figure 38.2: Notation for proof of the Pythagorean Theorem.Corollary 38.2 Then h = √ pq (see figure 38.2).Proof. (Exercise.)Corollary 38.3 a = √ qc and b = √ pc.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.