(1)证明:∵AD∥BC, ∴∠DEC=∠EDA,∠BEA=∠EAD, 又∵EA=ED, ∴∠EAD=∠EDA, ∴∠DEC=∠AEB, 又∵EB=EC, ∴△DEC≌△AEB, ∴AB=CD, ∴梯形ABCD是等腰梯形.
(2)当AB⊥AC时,四边形AECD是菱形. 证明:∵AD∥BC,BE=EC=AD, ∴四边形ABED和四边形AECD均为平行四边形. ∴ABED, ∵AB⊥AC, ∴AE=BE=EC, ∴平行四边形AECD是菱形. 过A作AG⊥BE于点G, ∵AE=BE=AB=2, ∴△ABE是等边三角形, ∴∠AEB=60°, ∴AG=, ∴S菱形AECD=EC•AG=2×=2. |