( U , P ) ▁are ▁the ▁following : ▁▁ ▁Thom son ▁cub ic : ▁Z ( X ( 2 ), X ( 1 )), ▁where ▁X ( 2 ) ▁= ▁cent roid , ▁X ( 1 ) ▁= ▁in center ▁ ▁Feuer bach ▁cub ic : ▁Z ( X ( 5 ), X ( 1 )), ▁where ▁X ( 5 ) ▁= ▁Feuer bach ▁point ▁ ▁Dar b oux ▁cub ic : ▁Z ( X ( 2 0 ), X ( 1 )), ▁where ▁X ( 2 0 ) ▁= ▁De ▁Long ch amps ▁point ▁ ▁Ne uber g ▁cub ic : ▁Z ( X ( 3 0 ), X ( 1 )), ▁where ▁X ( 3 0 ) ▁= ▁E uler ▁infinity ▁point . ▁ ▁Con versions ▁ ▁Between ▁tr il ine ar ▁coordinates ▁and ▁distances ▁from ▁s idel ines ▁ ▁For ▁any ▁choice ▁of ▁tr il ine ar ▁coordinates ▁x : y : z ▁to ▁locate ▁a ▁point , ▁the ▁actual ▁distances ▁of ▁the ▁point ▁from ▁the ▁s idel ines ▁are ▁given ▁by ▁a ' ▁= ▁k x , ▁b ' ▁= ▁k y , ▁c ' ▁= ▁k z ▁where ▁k ▁can ▁be ▁determined ▁by ▁the ▁formula ▁ ▁in ▁which ▁a , ▁b , ▁c ▁are ▁the ▁respective ▁s idel ength s ▁BC , ▁CA , ▁AB , ▁and ▁ ∆ ▁is ▁the ▁area ▁of ▁ABC . ▁ ▁Between ▁b ary cent ric ▁and ▁tr il ine ar ▁coordinates ▁A ▁point ▁with ▁tr il ine ar ▁coordinates ▁x ▁: