∵AD=AE AO⊥DE
∴∠BAO=∠CAO=1/2 ∠BAC
∵∠ABO=∠CBO=1/2 ∠ABC
∴O是△内角平分线的交点
∴∠ACO=∠BCO=1/2 ∠ACB
∵∠OBC+∠OCB+∠BOC=180°
∴∠BOC=180°-∠OBC-∠OCB
=180°-1/2(∠ABC+∠ACB)
=180°-1/2(180°-∠BAC)
=90°+1/2 ∠BAC
=90°+∠BAO
∵∠BDO=∠AOD+∠BAO=90° +∠BAO
∴∠BDO=∠BOC
∵∠DBO=∠OBC
∴△BOD ∽△BCO
∴BO/BC=BD/BO
∴BO²=BD*BC