▁of ▁a ▁right ▁triangle , ▁see ▁the ▁hyper bol ic ▁triangle ▁of ▁a ▁hyper bol ic ▁sector . ▁ ▁Special ▁right ▁tri angles ▁ ▁The ▁values ▁of ▁the ▁trig on ometric ▁functions ▁can ▁be ▁evaluated ▁exactly ▁for ▁certain ▁angles ▁using ▁right ▁tri angles ▁with ▁special ▁angles . ▁These ▁include ▁the ▁ 3 0 - 6 0 - 9 0 ▁triangle ▁which ▁can ▁be ▁used ▁to ▁evaluate ▁the ▁trig on ometric ▁functions ▁for ▁any ▁multiple ▁of ▁ π / 6 , ▁and ▁the ▁ 4 5 - 4 5 - 9 0 ▁triangle ▁which ▁can ▁be ▁used ▁to ▁evaluate ▁the ▁trig on ometric ▁functions ▁for ▁any ▁multiple ▁of ▁ π / 4 . ▁ ▁Ke pler ▁triangle ▁Let ▁H , ▁G , ▁and ▁A ▁be ▁the ▁harm onic ▁mean , ▁the ▁geometric ▁mean , ▁and ▁the ▁arithmetic ▁mean ▁of ▁two ▁positive ▁numbers ▁a ▁and ▁b ▁with ▁a ▁> ▁b . ▁If ▁a ▁right ▁triangle ▁has ▁legs ▁H ▁and ▁G ▁and ▁hyp oten use ▁A , ▁then ▁ ▁and ▁ ▁where ▁ ▁is ▁the ▁golden ▁ratio ▁ ▁Since ▁the ▁sides ▁of ▁this ▁right ▁triangle ▁are ▁in ▁geometric ▁pro gression , ▁this ▁is ▁the ▁Ke pler ▁triangle . ▁ ▁Th ales ' ▁theorem ▁ ▁Th ales ' ▁theorem ▁states ▁that ▁if ▁A ▁is ▁any ▁point ▁of ▁the ▁circle ▁with ▁diameter ▁BC ▁( except ▁B ▁or ▁C ▁themselves ) ▁ABC ▁is ▁a ▁right ▁triangle ▁where ▁A ▁is ▁the ▁right ▁angle . ▁The ▁con verse ▁states ▁that ▁if ▁a ▁right ▁triangle ▁is ▁ins cribed ▁in ▁a ▁circle ▁then