如图, △ABC中, AB = AC, E在BC上, D在AE上. 则下列说法中正确的有( )
①若E为BC中点, 则有BD = CD; ②若BD = CD, 则E为BC中点;
③若AE⊥BC, 则有BD = CD; ④若BD = CD, 则AE⊥BC.
A. ①③④ B. ②③④ C. ①②③ D. ①②③④
【答案】D
【解析】①若E为BC中点, 则三线合一，AE⊥BC，所以BD = CD; ②若BD = CD,可证得∆ABD≅∆ACD,继而得到,所以AE是的角平分线，BC的垂直平分线，故E是BC的中点；③若AE⊥BC, 则三线合一，AE是BC的垂直平分线，得到BD=CD; ④若BD = CD, 则可证得∆ABD≅∆ACD,继而得到,所以AE是的角平分线，BC的垂直平分线，故AE⊥BC. 所以选D.