▁which ▁in ▁purely ▁relative ▁distances ▁simpl ifies ▁to ▁ ▁Form ulas ▁ ▁Col linear ities ▁and ▁conc urr encies ▁ ▁Tr il ine ar ▁coordinates ▁enable ▁many ▁algebraic ▁methods ▁in ▁triangle ▁geometry . ▁ ▁For ▁example , ▁three ▁points ▁ ▁P ▁= ▁p ▁: ▁q ▁: ▁r ▁U ▁= ▁u ▁: ▁v ▁: ▁w ▁X ▁= ▁x ▁: ▁y ▁: ▁z ▁ ▁are ▁col linear ▁if ▁and ▁only ▁if ▁the ▁determin ant ▁ ▁equals ▁zero . ▁ ▁Thus ▁if ▁x : y : z ▁is ▁a ▁variable ▁point , ▁the ▁equation ▁of ▁a ▁line ▁through ▁the ▁points ▁P ▁and ▁U ▁is ▁D ▁= ▁ 0 . ▁From ▁this , ▁every ▁straight ▁line ▁has ▁a ▁linear ▁equation ▁hom ogeneous ▁in ▁x , ▁y , ▁z . ▁ ▁Every ▁equation ▁of ▁the ▁form ▁l x + my + n z ▁= ▁ 0 ▁in ▁real ▁coefficients ▁is ▁a ▁real ▁straight ▁line ▁of ▁finite ▁points ▁unless ▁l ▁: ▁m : ▁n ▁is ▁proportional ▁to ▁a ▁: ▁b ▁: ▁c , ▁the ▁side ▁lengths , ▁in ▁which ▁case ▁we ▁have ▁the ▁loc us ▁of ▁points ▁at ▁infinity . ▁ ▁The ▁dual ▁of ▁this ▁proposition ▁is ▁that ▁the ▁lines ▁ ▁p α ▁+ ▁q β ▁+ ▁r γ ▁= ▁ 0 ▁u α ▁+ ▁v β ▁+ ▁w γ ▁= ▁ 0 , ▁x α ▁+ ▁y β ▁+ ▁z γ ▁= ▁ 0 ▁ ▁con cur ▁in ▁a ▁point ▁( α , ▁ β , ▁ γ ) ▁if ▁and ▁only ▁if ▁D ▁= ▁ 0 .