▁A ▁and ▁B ▁respectively ; ▁the ▁ratio ▁y : z ▁is ▁the ▁ratio ▁of ▁the ▁per pend icular ▁distances ▁from ▁the ▁point ▁to ▁the ▁s idel ines ▁opposite ▁vertices ▁B ▁and ▁C ▁respectively ; ▁and ▁like wise ▁for ▁z : x ▁and ▁vertices ▁C ▁and ▁A . ▁ ▁In ▁the ▁diagram ▁at ▁right , ▁the ▁tr il ine ar ▁coordinates ▁of ▁the ▁indicated ▁interior ▁point ▁are ▁the ▁actual ▁distances ▁( a ' ▁, ▁b ' ▁, ▁c ' ▁), ▁or ▁equival ently ▁in ▁ratio ▁form , ▁ka ' ▁: kb ' ▁: kc ' ▁ ▁for ▁any ▁positive ▁constant ▁k . ▁If ▁a ▁point ▁is ▁on ▁a ▁sid eline ▁of ▁the ▁reference ▁triangle , ▁its ▁corresponding ▁tr il ine ar ▁coordinate ▁is ▁ 0 . ▁If ▁an ▁exterior ▁point ▁is ▁on ▁the ▁opposite ▁side ▁of ▁a ▁sid eline ▁from ▁the ▁interior ▁of ▁the ▁triangle , ▁its ▁tr il ine ar ▁coordinate ▁associated ▁with ▁that ▁sid eline ▁is ▁negative . ▁It ▁is ▁impossible ▁for ▁all ▁three ▁tr il ine ar ▁coordinates ▁to ▁be ▁non - pos itive . ▁ ▁The ▁name ▁" tr il ine ar ▁coordinates " ▁is ▁sometimes ▁abbre vi ated ▁to ▁" tr il ine ars ". ▁ ▁Not ation ▁ ▁The ▁ratio ▁notation ▁x : y : z ▁for ▁tr il ine ar ▁coordinates ▁is ▁different ▁from ▁the ▁ordered ▁triple ▁notation ▁( a ' ▁, ▁b ' ▁, ▁c ' ▁) ▁for ▁actual ▁directed ▁distances . ▁Here ▁each ▁of ▁x , ▁y , ▁and ▁z ▁has ▁no ▁meaning ▁by ▁itself