theorem
  A,B,C is_a_triangle implies
  the_radius_of_the_circumcircle(A,B,C) = |.the_circumcenter(A,B,C)-B.| &
  the_radius_of_the_circumcircle(A,B,C) = |.the_circumcenter(A,B,C)-C.|
  proof
    assume
A1: A,B,C is_a_triangle;
    then |.the_circumcenter(A,B,C)-A.| = |.the_circumcenter(A,B,C)-B.| &
    |.the_circumcenter(A,B,C)-A.| = |.the_circumcenter(A,B,C)-C.| by Th50;
    hence thesis by A1,Def4;
  end;
