⑴Proof: ∵ ,

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∵ ,

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⑵The answer is not unique. like .

Proof: ∵ , ,

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?The similarity ratio is: ． ?

⑶ From (2), we get , . ??

Similarly.

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⑷ ,

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(1) Since △ABC is an equilateral triangle, we get AB=BC, ∠BAC=∠BCA=60°, and since the quadrilateral ACDE is an isosceles trapezoid, we get AE =CD, ∠ACD=∠CAE=60°, use "SAS" to determine △ABE≌△CBD;

(2) Existence. Similar triangles can be obtained by using AB∥CD or AE∥BC;

(3) can be obtained from the conclusion of (2), that is, in the same way, AM=AC can be obtained, and AM=MN=NC can be proved;

(4) Draw the extension line of DF⊥BC intersecting BC at F. In Rt△CDF, from ∠CDF=30°, CD=AE=1, we can find CF, DF, in Rt△ In BDF, find BD according to the Pythagorean theorem.