如图,在$\triangle ABC$中,$AB=AC$,过点$A$作$AD\bot BC$交$BC$于点$D$.点$E$是线段$AD$上一点,连接$BE$,请完成下面的作图和填空.
$(1)$用尺规完成以下基本作图:以点$C$为顶点,在$BC$的右边作$\angle BCF=\angle EBD$,射线$CF$交$AD$的延长线于点$F$,连接$BF$,$FC.(保留作图痕迹,不写作法,不下结论)$
$(2)$求证:四边形$BECF$是菱形.
证明:$\because AB=AC$,$AD\bot BC$,
$\therefore$______,
$\therefore BE=CE$.
在$\triangle BED$和$\triangle CFD$中,$\left\{\begin{array}{l}∠EBD=∠CFD\\ BD=DC\\(ㅤㅤ)\end{array}\right.$,______
$\therefore \triangle BED$≌$\triangle CFD$,
$\therefore BE=CF$.
$\because \angle EBD=\angle BCF$,
$\therefore$______,
$\therefore$四边形$BECF$是平行四边形.
$\because$______,
$\therefore$四边形$BECF$是菱形.