▁y ▁: ▁z ▁has ▁b ary cent ric ▁coordinates ▁ax ▁: ▁by ▁: ▁cz ▁where ▁a , ▁b , ▁c ▁are ▁the ▁s idel ength s ▁of ▁the ▁triangle . ▁ ▁Con vers ely , ▁a ▁point ▁with ▁b ary cent rics ▁α ▁: ▁ β ▁: ▁ γ ▁has ▁tr il ine ar ▁coordinates ▁α / a ▁: ▁ β / b ▁: ▁ γ / c . ▁ ▁Between ▁Cart esian ▁and ▁tr il ine ar ▁coordinates ▁Given ▁a ▁reference ▁triangle ▁ABC , ▁express ▁the ▁position ▁of ▁the ▁vertex ▁B ▁in ▁terms ▁of ▁an ▁ordered ▁pair ▁of ▁Cart esian ▁coordinates ▁and ▁represent ▁this ▁algebra ically ▁as ▁a ▁vector ▁B , ▁using ▁vertex ▁C ▁as ▁the ▁origin . ▁Similarly ▁define ▁the ▁position ▁vector ▁of ▁vertex ▁A ▁as ▁A . ▁Then ▁any ▁point ▁P ▁associated ▁with ▁the ▁reference ▁triangle ▁ABC ▁can ▁be ▁defined ▁in ▁a ▁Cart esian ▁system ▁as ▁a ▁vector ▁P ▁= ▁k 1 A ▁+ ▁k 2 B . ▁If ▁this ▁point ▁P ▁has ▁tr il ine ar ▁coordinates ▁x ▁: ▁y ▁: ▁z ▁then ▁the ▁conversion ▁formula ▁from ▁the ▁coefficients ▁k 1 ▁and ▁k 2 ▁in ▁the ▁Cart esian ▁representation ▁to ▁the ▁tr il ine ar ▁coordinates ▁is , ▁for ▁side ▁lengths ▁a , ▁b , ▁c ▁opposite ▁vertices ▁A , ▁B , ▁C , ▁▁▁▁ ▁and ▁the ▁conversion ▁formula ▁from ▁the ▁tr il ine ar ▁coordinates ▁to ▁the ▁coefficients ▁in ▁the ▁Cart esian ▁representation ▁is ▁▁▁▁ ▁More ▁generally , ▁if ▁an ▁arbitrary ▁origin ▁is ▁chosen ▁where ▁the ▁Cart esian