$证明：(1)在△BOD和△COE中$
$\begin{cases}{∠BOD=∠COE}\\{∠B=∠C}\\{BD=CE}\end{cases}$
$∴△BOD≌△COE(\mathrm {AAS})$
$∴OD= OE$
$(2)∵点D，E分别是AB，AC的中点$
$∴AD= BD=\frac {1}{2}AB，AE=CE=\frac {1}{2}AC$
$∵BD= CE$
$∴AD=AE，AB=AC$
$在△ABE和△ACD中$
$\begin{cases}{AB=AC }\\{∠A=∠A} \\{AE=AD} \end{cases}$
$∴△ABE≌△ACD(\mathrm {SAS})$
