Plane Geometry - Bruce E. Shapiro

Section 35Similar TrianglesLet us recall the definition of similar triangles, previously stated as definition32.12:Definition 35.1 (Similar Triangles) Triangles △ABC and △DEF aresaid to be similar if ∠ABC ∼ = ∠DEF , ∠BCA ∼ = ∠EF D, ∠CAB ∼ =∠F DE, and we write △ABC ∼ △DEF .Theorem 35.2 (Fundamental Theorem on Similar Triangles)△ABC ∼ △DEF ⇒ ABAC = DEDFProof. Let △ABC ∼ △DEF .If AB = DE then △ABC ∼ = △DEF by ASA, and hence AC = DF by theproperties of congruent triangles, so that AB/AC = DE/DF .Suppose that AB ≠ DE. Then either AB > DE or AB < DE.Suppose that AB > DE. Choose B ′ ∈ AB such that AB ′ = DE.Construct a line m through B ′ that is parallel to ←→ BC. By Pasch’s theorem(theorem 15.12), m intersects AC. Call the point of intersection C ′ (seefigure 35.1).By the corresponding angles theorem η = β = ɛ; hence by ASA (α = δ,AB ′ = DE, and η = ɛ), △AB ′ C ′ ∼ = △DEF . Hence AC ′ = DF .Construct line n through A such that n ‖ m ‖ ←→ BC.177