Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.
Use parallel segments to derive proportional sides
Problem
In triangle \(ABC\), \(DE~\parallel~BC\) with \(AD~=~4\), \(DB~=~6\), and \(AE~=~5\). Establish \(\triangle~ADE~∼~\triangle~ABC\) by \(AA\), solve \(AD/DB~=~AE/EC\) for \(EC\), then verify the equal whole-side ratios \(AD/AB\) and \(AE/AC\).
Parallelism connects two complementary views of this diagram: AA similarity and proportional side splitting. We’ll establish the correspondence, solve an upper-to-lower segment proportion, rebuild both whole sides, and verify that the small-to-large ratios agree while keeping the two scale-factor directions distinct.
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