①∵BD为△ABC的
角平分线,∴∠ABD=∠CBD,
∴在△ABD和△EBC中,
,
∴△ABD≌△EBC(SAS),…①正确;
②∵BD为△ABC的角平分线,BD=BC,BE=BA,
∴∠BCD=∠BDC=∠BAE=∠BEA,
∵△ABD≌△EBC,∴∠BCE=∠BDA,
∴∠BCE+∠BCD=∠BDA+∠BDC=180° …②正确;
③∵∠BCE=∠BDA,∠BCE=∠BCD+∠DCE,∠BDA=∠DAE+∠BEA,∠BCD=∠BEA,
∴∠DCE=∠DAE,
∴△ACE为
等腰三角形∴AE=EC,
∵△ABD≌△EBC
∴AD=EC,
∴AD=AE=EC.…③正确;
④过E作EG⊥BC于G点,
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/5243fbf2b21193136a1f424a66380cd790238de8?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
∵E是BD上的点,∴EF=EG,
∵在RT△BEG和RT△BEF中,
,
∴RT△BEG≌RT△BEF(HL),
∴BG=BF,
∵在RT△CEG和RT△AFE中,
,
∴RT△CEG≌RT△AFE(HL),
∴AF=CG,
∴BA+BC=BF+FA+BG-CG=BF+BG=2BF.…④正确.
故选D.