Plane Geometry - Bruce E. Shapiro

178 SECTION 35. SIMILAR TRIANGLESBy the parallel projection theorem (theorem 34.2),AB ′AB = AC′ACSubstituting AC ′ = DF and AB ′ = DE,DEAB = DFACCross multiplying gives the result of the theorem.If AB < DE, then relabel the two triangles (exchange all the labels onthe first triangle with the corresponding item on the second triangle); thenrepeat the above argument to complete the proof.Figure 35.1: The fundamental theorem on similar triangles (theorem 35.2)tells us that AB/AC = DE/DF .Corollary 35.3 If △ABC ∼ △DEF then there is a number r > 0 suchthatDE = r × ABDF = r × ACEF = r × BCThe number r is called the common ratio of the sides of the triangle.Proof. Suppose that △ABC ∼ △DEF . Then by the fundamental theoremon similar triangles,DefineABAC = DEDFABandBC = DEEFr = DEAB« CC BY-NC-ND 3.0. Revised: 18 Nov 2012