▁of ▁this ▁equation ▁in ▁integer ▁values ▁of ▁a , ▁b , ▁f , ▁and ▁c , ▁see ▁here . ▁ ▁The ▁alt itude ▁from ▁either ▁leg ▁coinc ides ▁with ▁the ▁other ▁leg . ▁Since ▁these ▁intersect ▁at ▁the ▁right - ang led ▁vertex , ▁the ▁right ▁triangle ' s ▁orth oc enter — the ▁intersection ▁of ▁its ▁three ▁alt itudes — co inc ides ▁with ▁the ▁right - ang led ▁vertex . ▁ ▁P yth ag ore an ▁theorem ▁ ▁The ▁P yth ag ore an ▁theorem ▁states ▁that : ▁In ▁any ▁right ▁triangle , ▁the ▁area ▁of ▁the ▁square ▁whose ▁side ▁is ▁the ▁hyp oten use ▁( the ▁side ▁opposite ▁the ▁right ▁angle ) ▁is ▁equal ▁to ▁the ▁sum ▁of ▁the ▁areas ▁of ▁the ▁squares ▁whose ▁sides ▁are ▁the ▁two ▁legs ▁( the ▁two ▁sides ▁that ▁meet ▁at ▁a ▁right ▁angle ). ▁ ▁This ▁can ▁be ▁stated ▁in ▁equation ▁form ▁as ▁ ▁where ▁c ▁is ▁the ▁length ▁of ▁the ▁hyp oten use , ▁and ▁a ▁and ▁b ▁are ▁the ▁lengths ▁of ▁the ▁remaining ▁two ▁sides . ▁ ▁P yth ag ore an ▁tri ples ▁are ▁integer ▁values ▁of ▁a , ▁b , ▁c ▁satisfying ▁this ▁equation . ▁ ▁In radius ▁and ▁circum radius ▁ ▁The ▁radius ▁of ▁the ▁inc irc le ▁of ▁a ▁right ▁triangle ▁with ▁legs ▁a ▁and ▁b ▁and ▁hyp oten use ▁c ▁is ▁ ▁The ▁radius ▁of ▁the ▁circum circle ▁is ▁half ▁the ▁length ▁of ▁the ▁hyp oten use , ▁ ▁Thus ▁the ▁sum ▁of ▁the ▁circum radius ▁and