▁ ▁Also , ▁if ▁the ▁actual ▁directed ▁distances ▁are ▁used ▁when ▁evalu ating ▁the ▁determin ant ▁of ▁D , ▁then ▁the ▁area ▁of ▁triangle ▁P U X ▁is ▁K D , ▁where ▁K ▁= ▁abc / 8 ∆ 2 ▁( and ▁where ▁ ∆ ▁is ▁the ▁area ▁of ▁triangle ▁ABC , ▁as ▁above ) ▁if ▁triangle ▁P U X ▁has ▁the ▁same ▁orientation ▁( clock wise ▁or ▁counter clock wise ) ▁as ▁triangle ▁ABC , ▁and ▁K ▁= ▁– abc / 8 ∆ 2 ▁otherwise . ▁ ▁Par allel ▁lines ▁ ▁Two ▁lines ▁with ▁tr il ine ar ▁equations ▁ ▁and ▁ ▁are ▁parallel ▁if ▁and ▁only ▁if ▁ ▁where ▁a , ▁b , ▁c ▁are ▁the ▁side ▁lengths . ▁ ▁Ang le ▁between ▁two ▁lines ▁ ▁The ▁tang ents ▁of ▁the ▁angles ▁between ▁two ▁lines ▁with ▁tr il ine ar ▁equations ▁ ▁and ▁ ▁are ▁given ▁by ▁ ▁Per pend icular ▁lines ▁ ▁Thus ▁two ▁lines ▁with ▁tr il ine ar ▁equations ▁ ▁and ▁ ▁are ▁per pend icular ▁if ▁and ▁only ▁if ▁ ▁Alt itude ▁ ▁The ▁equation ▁of ▁the ▁alt itude ▁from ▁vertex ▁A ▁to ▁side ▁BC ▁is ▁ ▁Line ▁in ▁terms ▁of ▁distances ▁from ▁vertices ▁ ▁The ▁equation ▁of ▁a ▁line ▁with ▁variable ▁distances ▁p , ▁q , ▁r ▁from ▁the ▁vertices ▁A , ▁B , ▁C ▁whose ▁opposite ▁sides ▁are ▁a , ▁b , ▁c ▁is ▁ ▁Act ual - distance ▁tr il ine ar ▁coordinates ▁ ▁The ▁tr il ine ars ▁with ▁the ▁coordinate ▁values ▁a ',