19.「2026广东茂名期中，★★☆」(8分)如图,在四边形ABCD中,AB//DC,AB=AD,对角线AC,BD交于点O,AC平分∠BAD,过点C作CE⊥AB交AB的延长线于点E,连接OE.
(1)求证:四边形ABCD是菱形.
(2)若$AB=\sqrt{5},BD=2,$求OE的长.

19.解析
(1)证明:$\because AB// DC$,$\therefore ∠ OAB=∠ DCA$, ...... (1分)
$\because AC$平分$∠ DAB$,$\therefore ∠ OAB=∠ DAC$, ............ (2分)
$\therefore ∠ DCA=∠ DAC$,$\therefore CD=AD=AB$, ............ (3分)
又$\because AB// DC$,$\therefore$ 四边形ABCD是菱形. ............ (4分)
(2)$\because$ 四边形ABCD是菱形,
$\therefore OA=OC$,$OB=\frac{1}{2}BD=1$,$BD⊥ AC$,
$\because CE⊥ AB$,$\therefore OE=\frac{1}{2}AC=OA$, ............ (6分)
在$\mathrm{Rt}△ AOB$中,$AB=\sqrt{5}$,$OB=1$,
$\therefore OA=\sqrt{AB^2-OB^2}=\sqrt{5-1}=2$, ............ (7分)
$\therefore OE=OA=2$. ............ (8分)