irc le ▁ ▁The ▁triangle ▁can ▁be ▁ins cribed ▁in ▁a ▁sem ic irc le , ▁with ▁one ▁side ▁coinc iding ▁with ▁the ▁entire ty ▁of ▁the ▁diameter ▁( Th ales ' ▁theorem ). ▁ ▁The ▁circum center ▁is ▁the ▁mid point ▁of ▁the ▁longest ▁side . ▁ ▁The ▁longest ▁side ▁is ▁a ▁diameter ▁of ▁the ▁circum circle ▁▁ ▁The ▁circum circle ▁is ▁tang ent ▁to ▁the ▁nine - point ▁circle . ▁ ▁The ▁orth oc enter ▁lies ▁on ▁the ▁circum circle . ▁ ▁The ▁distance ▁between ▁the ▁in center ▁and ▁the ▁orth oc enter ▁is ▁equal ▁to ▁. ▁ ▁Tr ig on ometric ▁r ati os ▁ ▁The ▁trig on ometric ▁functions ▁for ▁ac ute ▁angles ▁can ▁be ▁defined ▁as ▁r ati os ▁of ▁the ▁sides ▁of ▁a ▁right ▁triangle . ▁For ▁a ▁given ▁angle , ▁a ▁right ▁triangle ▁may ▁be ▁constructed ▁with ▁this ▁angle , ▁and ▁the ▁sides ▁l abeled ▁opposite , ▁adjacent ▁and ▁hyp oten use ▁with ▁reference ▁to ▁this ▁angle ▁according ▁to ▁the ▁definitions ▁above . ▁These ▁r ati os ▁of ▁the ▁sides ▁do ▁not ▁depend ▁on ▁the ▁particular ▁right ▁triangle ▁chosen , ▁but ▁only ▁on ▁the ▁given ▁angle , ▁since ▁all ▁tri angles ▁constructed ▁this ▁way ▁are ▁similar . ▁If , ▁for ▁a ▁given ▁angle ▁α , ▁the ▁opposite ▁side , ▁adjacent ▁side ▁and ▁hyp oten use ▁are ▁l abeled ▁O , ▁A ▁and ▁H ▁respectively , ▁then ▁the ▁trig on ometric ▁functions ▁are ▁ ▁For ▁the ▁expression ▁of ▁hyper bol ic ▁functions ▁as ▁ratio ▁of ▁the ▁sides