∠BIC = 180◦− ∠IBC − ∠ICB = 180◦− 0.5∠ABC − 0.5∠ACB
= 180◦− 0.5(∠ABC + ∠ACB) = 180◦− 0.5(180◦− ∠BAC) = 90◦
+ 0.5∠BAC

∠BAC < 60◦
2∠BAC < 90◦+0.5∠BAC,
=∠BOC < ∠BIC

∠BPC < ∠BOC = 2∠BAC < 120◦

∠BPC = 180◦− (∠PBC + ∠PCB) < 180◦−(∠B + ∠C)/2
= 90 + ∠A/2< 120◦