Explanation:
You can use the definition of similarity to show that corresponding sides are proportional. You cannot prove anything about the proportionality of non-corresponding sides.
Statement . . . . Reason
1. BD║CE . . . . given
2. ∠A≅∠A . . . . reflexive property of congruence
3. ∠ABD≅∠ACE . . . . corresponding angles theorem
4. ΔABD ~ ΔACE . . . . AA similarity postulate
5. BD/CE = AD/AE . . . . definition of similarity
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Additional comment
Length DE is related to BC in the same way that AD is related to AB. There is no way to prove the ratio you show in your problem statement. You can show that ...
DE/AE = (CE -BD)/CE = 1 -(BD/CE) . . . . the difference is proportional to the corresponding longer side
