AX = AB + BC − CA
----------------------
2
AY = BC + CA − AB
----------------------
2
AD = AE = CA + AB − BC
-----------------------------
2
BD = AB − AD = AB − CA + AB − BC
------------------------------------------
2
= AB + BC − CA
------------------
2
= AX.

~~~~~~~~~這是分隔線~~~~~~~~~~~

∠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

-----------------------------------
到這裡OK,
但為什麼∠BPC < ∠BOC ? ∠PBC + ∠PCB> (∠B + ∠C)/2?

若∠A最小,則BC最短,證明很容易.

但若AB<BC,
則AY=(BC+CA-AB)/2 > AC/2=AN,
圖就不一樣了,
也看不出∠BPC < ∠BOC了.