▁the ▁in radius ▁is ▁half ▁the ▁sum ▁of ▁the ▁legs : ▁ ▁One ▁of ▁the ▁legs ▁can ▁be ▁expressed ▁in ▁terms ▁of ▁the ▁in radius ▁and ▁the ▁other ▁leg ▁as ▁ ▁Character izations ▁A ▁triangle ▁ABC ▁with ▁sides ▁, ▁semi per imeter ▁s , ▁area ▁T , ▁alt itude ▁h ▁opposite ▁the ▁longest ▁side , ▁circum radius ▁R , ▁in radius ▁r , ▁ex rad ii ▁ra , ▁r b , ▁r c ▁( t ang ent ▁to ▁a , ▁b , ▁c ▁respectively ), ▁and ▁med ians ▁ma , ▁m b , ▁m c ▁is ▁a ▁right ▁triangle ▁if ▁and ▁only ▁if ▁any ▁one ▁of ▁the ▁statements ▁in ▁the ▁following ▁six ▁categories ▁is ▁true . ▁All ▁of ▁them ▁are ▁of ▁course ▁also ▁properties ▁of ▁a ▁right ▁triangle , ▁since ▁character izations ▁are ▁equival ences . ▁ ▁S ides ▁and ▁semi per imeter ▁ ▁Ang les ▁ ▁A ▁and ▁B ▁are ▁complement ary . ▁ ▁Area ▁▁▁▁▁▁▁▁ ▁where ▁P ▁is ▁the ▁tang ency ▁point ▁of ▁the ▁inc irc le ▁at ▁the ▁longest ▁side ▁AB . ▁ ▁In radius ▁and ▁ex rad ii ▁ ▁Alt itude ▁and ▁med ians ▁▁▁▁▁▁ ▁The ▁length ▁of ▁one ▁median ▁is ▁equal ▁to ▁the ▁circum radius . ▁ ▁The ▁short est ▁alt itude ▁( the ▁one ▁from ▁the ▁vertex ▁with ▁the ▁biggest ▁angle ) ▁is ▁the ▁geometric ▁mean ▁of ▁the ▁line ▁segments ▁it ▁div ides ▁the ▁opposite ▁( long est ) ▁side ▁into . ▁This ▁is ▁the ▁right ▁triangle ▁alt itude ▁theorem . ▁ ▁Circ um circle ▁and ▁inc