(2013•黄浦区二模)如图,在梯形ABCD中,AD∥BC,AB=CD,对角线AC与BD交于点O,OE⊥BC,垂足是E.
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![](http://img.wesiedu.com/upload/f/90/f909b07f64abc643e5a2437c57f9cccf.jpg)
(1)求证:E是BC的中点;
(2)若在线段BO上存在点P,使得四边形AOEP为平行四边形.求证:四边形ABED是平行四边形.
![(2013•黄浦区二模)如图,在梯形ABCD中,AD∥BC,AB=CD,对角线AC与BD交于点O,OE⊥BC,垂足是E.](/uploads/image/z/633859-43-9.jpg?t=%EF%BC%882013%E2%80%A2%E9%BB%84%E6%B5%A6%E5%8C%BA%E4%BA%8C%E6%A8%A1%EF%BC%89%E5%A6%82%E5%9B%BE%EF%BC%8C%E5%9C%A8%E6%A2%AF%E5%BD%A2ABCD%E4%B8%AD%EF%BC%8CAD%E2%88%A5BC%EF%BC%8CAB%3DCD%EF%BC%8C%E5%AF%B9%E8%A7%92%E7%BA%BFAC%E4%B8%8EBD%E4%BA%A4%E4%BA%8E%E7%82%B9O%EF%BC%8COE%E2%8A%A5BC%EF%BC%8C%E5%9E%82%E8%B6%B3%E6%98%AFE%EF%BC%8E)
(1)证明:在梯形ABCD中,AD∥BC,AB=CD,
∵在△ABC和△DCB中,
AB=DC
AC=DB
BC=CB,
∴△ABC≌△DCB(SSS),
∴∠ACB=∠DBC,
∴OB=OC,
∵OE⊥BC,
∴点E是BC中点(三线合一).
(2)∵四边形AOEP是平行四边形,
∴AP=OE,
∵在△APD和△EOB中,
∠ADP=∠EBO
∠APD=∠EOB
AP=EO,
∴△APD≌△EOB(AAS),
∴AD=BE,
又∵AD∥BC,
∴四边形ABED是平行四边形.
∵在△ABC和△DCB中,
AB=DC
AC=DB
BC=CB,
∴△ABC≌△DCB(SSS),
∴∠ACB=∠DBC,
∴OB=OC,
∵OE⊥BC,
∴点E是BC中点(三线合一).
![](http://img.wesiedu.com/upload/c/a4/ca4a39ea25a7f1d3aac087fa749cbb57.jpg)
∴AP=OE,
∵在△APD和△EOB中,
∠ADP=∠EBO
∠APD=∠EOB
AP=EO,
∴△APD≌△EOB(AAS),
∴AD=BE,
又∵AD∥BC,
∴四边形ABED是平行四边形.
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