c ▁= ▁c sc ▁A ▁: ▁c sc ▁B ▁: ▁c sc ▁C . ▁ ▁circum center ▁= ▁cos ▁A ▁: ▁cos ▁B ▁: ▁cos ▁C . ▁ ▁orth oc enter ▁= ▁sec ▁A ▁: ▁sec ▁B ▁: ▁sec ▁C . ▁ ▁nine - point ▁center ▁= ▁cos ( B ▁− ▁C ) ▁: ▁cos ( C ▁− ▁A ) ▁: ▁cos ( A ▁− ▁B ). ▁ ▁sym med ian ▁point ▁= ▁a ▁: ▁b ▁: ▁c ▁= ▁sin ▁A ▁: ▁sin ▁B ▁: ▁sin ▁C . ▁ ▁A - ex center ▁= ▁− 1 ▁: ▁ 1 ▁: ▁ 1 ▁ ▁B - ex center ▁= ▁ 1 ▁: ▁− 1 ▁: ▁ 1 ▁ ▁C - ex center ▁= ▁ 1 ▁: ▁ 1 ▁: ▁− 1 . ▁ ▁Note ▁that , ▁in ▁general , ▁the ▁in center ▁is ▁not ▁the ▁same ▁as ▁the ▁cent roid ; ▁the ▁cent roid ▁has ▁b ary cent ric ▁coordinates ▁ 1 ▁: ▁ 1 ▁: ▁ 1 ▁( th ese ▁being ▁proportional ▁to ▁actual ▁signed ▁areas ▁of ▁the ▁tri angles ▁B GC , ▁CG A , ▁A GB , ▁where ▁G ▁= ▁cent roid .) ▁ ▁The ▁mid point ▁of , ▁for ▁example , ▁side ▁BC ▁has ▁tr il ine ar ▁coordinates ▁in ▁actual ▁sid eline ▁distances ▁▁ ▁for ▁triangle ▁area ▁, ▁which ▁in ▁arbitr arily ▁specified ▁relative ▁distances ▁simpl ifies ▁to ▁ ▁The ▁coordinates ▁in ▁actual ▁sid eline ▁distances ▁of ▁the ▁foot ▁of ▁the ▁alt itude ▁from ▁A ▁to ▁BC ▁are ▁