▁in ▁tr il ine ar ▁coordinates ▁x , ▁y , ▁z ▁of ▁any ▁circum con ic ▁of ▁a ▁triangle ▁is ▁ ▁If ▁the ▁parameters ▁l , ▁m , ▁n ▁respectively ▁equal ▁the ▁side ▁lengths ▁a , ▁b , ▁c ▁( or ▁the ▁s ines ▁of ▁the ▁angles ▁opposite ▁them ) ▁then ▁the ▁equation ▁gives ▁the ▁circum circle . ▁ ▁Each ▁distinct ▁circum con ic ▁has ▁a ▁center ▁unique ▁to ▁itself . ▁The ▁equation ▁in ▁tr il ine ar ▁coordinates ▁of ▁the ▁circum con ic ▁with ▁center ▁x ' ▁: ▁ ▁y ' ▁: ▁z ' ▁ ▁is ▁ ▁In con ics ▁ ▁Every ▁con ic ▁section ▁ins cribed ▁in ▁a ▁triangle ▁has ▁an ▁equation ▁in ▁tr il ine ar ▁coordinates : ▁ ▁with ▁exactly ▁one ▁or ▁three ▁of ▁the ▁un spec ified ▁signs ▁being ▁negative . ▁ ▁The ▁equation ▁of ▁the ▁inc irc le ▁can ▁be ▁simplified ▁to ▁ ▁while ▁the ▁equation ▁for , ▁for ▁example , ▁the ▁exc irc le ▁adjacent ▁to ▁the ▁side ▁segment ▁opposite ▁vertex ▁A ▁can ▁be ▁written ▁as ▁ ▁Cub ic ▁curves ▁Many ▁cub ic ▁curves ▁are ▁easily ▁represented ▁using ▁tr il ine ar ▁coordinates . ▁ ▁For ▁example , ▁the ▁p iv otal ▁self - iso con j ug ate ▁cub ic ▁Z ( U , P ), ▁as ▁the ▁loc us ▁of ▁a ▁point ▁X ▁such ▁that ▁the ▁P - iso con j ug ate ▁of ▁X ▁is ▁on ▁the ▁line ▁U X ▁is ▁given ▁by ▁the ▁determin ant ▁equation ▁ ▁Among ▁named ▁cub ics ▁Z