▁coordinates ▁of ▁the ▁vertices ▁are ▁known ▁and ▁represented ▁by ▁the ▁vectors ▁A , ▁B ▁and ▁C ▁and ▁if ▁the ▁point ▁P ▁has ▁tr il ine ar ▁coordinates ▁x ▁: ▁y ▁: ▁z , ▁then ▁the ▁Cart esian ▁coordinates ▁of ▁P ▁are ▁the ▁weight ed ▁average ▁of ▁the ▁Cart esian ▁coordinates ▁of ▁these ▁vertices ▁using ▁the ▁b ary cent ric ▁coordinates ▁ax , ▁by ▁and ▁cz ▁as ▁the ▁weights . ▁Hence ▁the ▁conversion ▁formula ▁from ▁the ▁tr il ine ar ▁coordinates ▁x , ▁y , ▁z ▁ ▁to ▁the ▁vector ▁of ▁Cart esian ▁coordinates ▁P ▁of ▁the ▁point ▁is ▁given ▁by ▁▁▁ ▁where ▁the ▁side ▁lengths ▁are ▁| C ▁− ▁B | ▁= ▁a , ▁| A ▁− ▁C | ▁= ▁b ▁and ▁| B ▁− ▁A | ▁= ▁c . ▁ ▁See ▁also ▁ ▁Mor ley ' s ▁tr is ector ▁theorem # Mor ley ' s ▁tri angles , ▁giving ▁examples ▁of ▁numerous ▁points ▁expressed ▁in ▁tr il ine ar ▁coordinates ▁T ern ary ▁plot ▁Viv iani ' s ▁theorem ▁ ▁References ▁ ▁External ▁links ▁ ▁Encyclopedia ▁of ▁Tri angle ▁Cent ers ▁- ▁E TC ▁by ▁Clark ▁Kim ber ling ; ▁has ▁tr il ine ar ▁coordinates ▁( and ▁b ary cent ric ) ▁for ▁more ▁than ▁ 7 0 0 0 ▁triangle ▁cent ers ▁ ▁Category : Linear ▁algebra ▁Category : Aff ine ▁geometry ▁Category : Tri angle ▁geometry ▁Category : Co ordinate ▁systems <0x0A> </s> ▁On ▁Language ▁was ▁a ▁regular ▁column ▁in ▁the ▁week ly ▁New ▁York ▁Times ▁Magazine ▁on